A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion.
Kuniba, Nakanishi, and Suzuki (1994) have formulated a general conjecture expressing the positive solution of an $\ell$-restricted $Q$-system in terms of quantum dimensions of Kirillov-Reshetikhin modules. After presenting this conjecture, we sketch a proof for the exceptional type $E_6$ following our preprint (2013). In types $E_7$ and $E_8$, we prove positivity for a subset of the nodes of the Dynkin diagram, and we reduce the positivity for the remaining nodes to the conjectural iterated log-concavity of certain explicit sequences of real algebraic numbers.
We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.
A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus.
The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. However, instead of requiring the tail of one permutation to equal the head of another for them to be connected by an edge, we require that the head and tail in question have their letters appear in the same order of size. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping $312$-avoiding permutations. Using this we also give a refinement of the enumeration of $312$-avoiding affine permutations.
Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for $T_X$ (ıe operators that do not annihilate $T_X$) and the space of nice differential operators (ıe operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the (arithmetic) matroid defined by $X$.
We consider a carries process which is a generalization of that by Holte in the sense that (i) we take various digit sets, and (ii) we also consider negative base. Our results are : (i) eigenvalues and eigenvectors of the transition probability matrices, and their connection to combinatorics and representation theory, (ii) an application to the computation of the distribution of the sum of i.i.d. uniform r.v.'s on [0,1], (iii) a relation to riffle shuffle.
We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra.
With a crystallographic root system $\Phi$ , there are associated two Catalan objects, the set of nonnesting partitions $NN(\Phi)$, and the cluster complex $\Delta (\Phi)$. These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton. We prove this conjecture, and indicate its generalisation for the Fuß-Catalan objects $NN^{(k)}(\Phi)$ and $\Delta^{(k)}(\Phi)$, conjectured by Armstrong.
\textbfAbstract. We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. \bigbreak
In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In fact, the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in Lorentz space, the set of limit directions of weights equals the set of limit roots.
We define a $0$-Hecke action on composition tableaux, and then use it to derive $0$-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions.
We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our method gives two simple conditions under which the characteristic polynomial factors. We will see that Stanley's Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate a new result in graph theory and give new proofs of some classic results concerning the Möbius function.
We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph.
We consider two aspects of Kronecker coefficients in the directions of representation theory and combinatorics. We consider a conjecture of Jan Saxl stating that the tensor square of the $S_n$-irreducible representation indexed by the staircase partition contains every irreducible representation of $S_n$. We present a sufficient condition allowing to determine whether an irreducible representation is a constituent of a tensor square and using this result together with some analytic statements on partitions we prove Saxl conjecture for several partition classes. We also use Kronecker coefficients to give a new proof and a generalization of the unimodality of Gaussian ($q$-binomial) coefficients as polynomials in $q$, and extend this to strict unimodality.
We present a very simple explicit technique to generate a large family of point configurations with neighborly Delaunay triangulations. This proves that there are superexponentially many combinatorially distinct neighborly $d$-polytopes with $n$ vertices that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (and thus also of Delaunay triangulations). It coincides with the current best lower bound for the number of combinatorial types of polytopes.
We consider the multivariate generating series $F_P$ of $P-$partitions in infinitely many variables $x_1, x_2, \ldots$ . For some family of ranked posets $P$, it is natural to consider an analog $N_P$ with two infinite alphabets. When we collapse these two alphabets, we trivially recover $F_P$. Our main result is the converse, that is, the explicit construction of a map sending back $F_P$ onto $N_P$. We also give a noncommutative analog of the latter. An application is the construction of a basis of $\mathbf{WQSym}$ with a non-negative multiplication table, which lifts a basis of $\textit{QSym}$ introduced by K. Luoto.
Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations: $k$-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analogue of the Bruhat order on them. We show that certain families of $k$-Dyck tilings are in bijection with intervals in this order. We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations.
Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots, n\}$. Moreover, they raised several conjectures on the purity for this symmetric relation, in particular, on the Boolean cube $2^{[n]}$. In 0909.1423[math.CO] we proved these purity conjectures for the Boolean cube $2^{[n]}$, the discrete Grassmanian $\binom{[n]}{r}$, and some other set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for weakly separated collections inside a positroid which contain a Grassmann necklace $\mathcal {N}$ defining the positroid. We denote such set-systems as $\mathcal{Int}(\mathcal {N} )$. In this paper we give an alternative (and shorter) proof of the purity of $\mathcal{Int}(\mathcal {N} )$ and present a stronger result. More precisely, we introduce a set-system $\mathcal{Out}(\mathcal {N} )$ complementary to $\mathcal{Int}(\mathcal {N })$, in a sense, and establish its purity. Moreover, we prove (Theorem~3) that these two set-systems are weakly separated from each other. As a consequence of Theorem~3, we obtain the purity of set-systems related to pairs of weakly separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise a conjecture on the purity of both the interior and exterior of a generalized necklace.
The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.
An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. J.-L. Loday gave a particularly elegant realization of the associahedron, which was then generalized in two directions: on the one hand to obtain realizations of graph associahedra, and on the other hand to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and extend these two constructions to signed tree associahedra.
This paper constructs a bijection between irreducible $k$-shapes and surjective pistols of height $k-1$, which carries the "free $k$-sites" to the fixed points of surjective pistols. The bijection confirms a conjecture of Hivert and Mallet (FPSAC 2011) that the number of irreducible $k$-shape is counted by the Genocchi number $G_{2k}$.
In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients.
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.
We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset $P$ as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When $P = [a] \times [b]$, a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. We prove some homomesy results, showing that for certain functions $f$, the average of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen.
We expose a rule for multiplying a general Schubert polynomial with a power sum polynomial in $k$ variables. A signed sum over cyclic permutations replaces the signed sum over rim hooks in the classical Murgnahan-Nakayama rule. In the intersection theory of flag manifolds this computes all intersections of Schubert cycles with tautological classes coming from the Chern character. We also discuss extensions of this rule to small quantum cohomology.
The $\textit{parallel chip-firing game}$ is an automaton on graphs in which vertices "fire'' chips to their neighbors. This simple model, analogous to sandpiles forming and collapsing, contains much emergent complexity and has connections to different areas of mathematics including self-organized criticality and the study of the sandpile group. In this work, we study $\textit{firing sequences}$, which describe each vertex's interaction with its neighbors in this game. Our main contribution is a complete characterization of the periodic firing sequences that can occur in a game, which have a surprisingly simple combinatorial description. We also obtain other results about local behavior of the game after introducing the concept of $\textit{motors}$.
In our previous works "Pfaffian decomposition and a Pfaffian analogue of $q$-Catalan Hankel determinants'' (by M.Ishikawa, H. Tagawa and J. Zeng, J. Combin. Theory Ser. A, 120, 2013, 1263-1284) we have proposed several ways to evaluate certain Catalan-Hankel Pffafians and also formulated several conjectures. In this work we propose a new approach to compute these Catalan-Hankel Pffafians using Selberg's integral as well as their $q$-analogues. In particular, this approach permits us to settle most of the conjectures in our previous paper.
We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions.
We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. The formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It gives by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. The formula is similar in look to the one discovered by Goulden and Jackson for triangulations (although the latter does not rely on an additional Tutte equation). Both of them have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved - should such an interpretation exist, the history of bijective methods for maps would tend to show that the case treated here is easier to start with than the one of triangulations.
At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all.
In this abstract, I will survey the story of two enumerative miracles that relate certain Coxeter-theoretic objects and other poset-theoretic objects. The first miracle relates reduced words and linear extensions, while the second may be thought of as relating group elements and order ideals. The purpose of this abstract is to use a conjecture from my thesis to present both miracles in the same light.
We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$. We show that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k] \times [n-k]$ of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). This shows the existence of a flag simplicial polytope whose Stanley-Reisner ideal is an initial ideal of the Graßmann-Plücker ideal, while previous constructions of such a polytope did not guaranteed flagness. The simplicial complex and the polytope derived from it naturally reflect the relations between Graßmannians with different parameters, in particular the isomorphism $G_{k,n} \cong G_{n-k,n}$. This simplicial complex is closely related to the weak separability complex introduced by Zelevinsky and Leclerc.
A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The Pell permutations are in bijection with combinatorial objects that we call sashes, that is, tilings of a 1 by n rectangle with three types of tiles: black 1 by 1 squares, white 1 by 1 squares, and white 1 by 2 rectangles. The bijection induces a Hopf algebra structure on sashes. We describe the product and coproduct in terms of sashes, and the natural partial order on sashes. We also describe the dual coproduct and dual product of the dual Hopf algebra of sashes.
For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the $\textit{order}$ of the first entries is independent of the $\textit{order}$ of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated.
Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne.
We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals $1/e^2$ asymptotically. We also prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or <i>positive MacPhersonian</i>) is homeomorphic to a closed ball.
Sweep maps are a family of maps on words that, while simple to define, are not yet known to be injective in general. This family subsumes many of the "zeta maps" that have arisen in the study of q,t-Catalan numbers in the course of relating the three statistics of area, bounce and dinv. A sweep map can be defined for words over arbitrary alphabets with arbitrary weights. The latter property makes them particularly suitable for the study of rational Catalan combinatorics.
We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.
We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.
There are numerous combinatorial objects associated to a Grassmannian permutation $w_λ$ that index cells of the totally nonnegative Grassmannian. We study some of these objects (rook placements, acyclic orientations, various restricted fillings) and their q-analogues in the case of permutations $\mathcal{w}$ that are not necessarily Grassmannian.
We show that an element $\mathcal{w}$ of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement $\mathcal{I} (\mathcal{w})$ associated to the inversion set of \mathcal{w} is inductively free, and the product $(d_1+1) ...(d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval [e,w]. We also use Peterson translation of coconvex sets to give a Shapiro-Steinberg-Kostant rule for the exponents of $\mathcal{w}$.
This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.
This paper considers the representation theory of towers of algebras of $\mathcal{J} -trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras $FQSym , PBT$ , and $NCSF$ respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions.
Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.
The study of rhomboid-shaped fully packed loop configurations (RFPLs) is inspired by the work of Fischer and Nadeau on triangular fully packed loop configurations (TFPLs). By using the same techniques as they did some nice combinatorics for RFPLs arise. To each RFPL and to each oriented RFPL a quadruple of binary words (α ,β ;γ ,δ ) – its so-called boundary – is assigned. There are necessary conditions for the boundary of an RFPL respectively an oriented RFPL. For instance, it has to fulfill the inequality $d(γ )+d(δ )\geq(α )+d(β )+\vert α \vert _0\vert β \vert _1$, where $\vert α \vert _i $ denotes the number of occurrences of $i=0,1$ in α and d(α ) denotes the number of inversions of α . Furthermore, the number of ordinary RFPLs with boundary (α ,β ;γ ,δ ) can be expressed in terms of oriented RFPLs with the same boundary. Finally, oriented RFPLs with boundary (α ,β ;γ ,δ ) such that $d(γ )+d(δ )=d(α )+d(β )+\vert α \vert _0\vert β \vert _1$ are considered. They are in bijection with rhomboid-shaped Knutson-Tao puzzles. Also, Littlewood-Richardson tableaux of defect d are defined. They can be understood as a generalization of Littlewood-Richardson tableaux. Those tableaux are in bijection with rhomboid-shaped Knutson-Tao puzzles.
We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.
The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side $q,t$-enumerates parking functions in the $n ×n$ lattice. The symmetric function side may be simply expressed as $∇ e_n$ , where $∇$ is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and $e_n$ is the elementary symmetric function. The combinatorial side has been extended to parking functions in the $m ×n$ lattice for coprime $m,n$ by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases $m=2$ and $n=2$ .
A <i>stabilized-interval-free </i> (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if $s_n$ denotes the number of SIF permutations on [n], $S(z)=1+\sum_{n\geq1} s_n z^n$, and $F(z)=1+\sum_{n\geq1} n! z^n$, then $F(z)= S(zF(z))$. This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the \emphchord-connected permutations on [n], counted by $\{c_n\}_{n\geq1}$, whose generating function satisfies $S(z)= C(zS(z))$. The expressions at hand have immediate probabilistic interpretations, via the celebrated <i>moment-cumulant formula </i>of Speicher, in the context of the <i>free probability theory </i>of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian.
We introduce a poset structure on the reduced galleries in a supersolvable arrangement of hyperplanes. In particular, for Coxeter groups of type A or B, we construct a poset of reduced words for the longest element whose Hasse diagram is the graph of reduced words. Using Rambau's Suspension Lemma, we show that these posets are homotopy equivalent to spheres. We furthermore conjecture that its intervals are either homotopy equivalent to spheres or are contractible. One may view this as a analogue of a result of Edelman and Walker on the homotopy type of intervals of a poset of chambers of a hyperplane arrangement.
We define and enumerate a new class of self-avoiding walks on the square lattice, which we call <i>weakly prudent bridges</i>. Their definition is inspired by two previously-considered classes of self-avoiding walks, and can be viewed as a combination of those two models. We consider several methods for recursively generating these objects, each with its own advantages and disadvantages, and use these methods to solve the generating function, obtain very long series, and randomly generate walks of arbitrary size. We find that the growth constant of these walks is approximately 2.58, which is larger than that of any previously-solved class of self-avoiding walks.
We study permutation patterns from an algebraic combinatorics point of view. Using analogues of the classical shuffle and infiltration products for word, we define two new Hopf algebras of permutations related to the notion of permutation pattern. We show several remarkable properties of permutation patterns functions, as well their occurrence in other domains.
Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.
Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.
Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques.
The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are Wilf-equivalent in the factor order on $\mathbb{P}^{\ast}$ then they are rearrangements of each other. We introduce the notion of strong Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly Wilf-equivalent then they are rearrangements of each other. We further conjecture that Wilf-equivalence implies strong Wilf-equivalence.
We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. In this paper we examine the combinatorics of the quasisymmetric hook Schur functions, providing analogues of the Robinson-Schensted-Knuth algorithm and a generalized Cauchy Identity.
We use a recently introduced combinatorial object, the $\textit{interval-poset}$, to describe two bijections on intervals of the Tamari lattice. Both bijections give a combinatorial proof of some previously known results. The first one is an inner bijection between Tamari intervals that exchanges the $\textit{initial rise}$ and $\textit{lower contacts}$ statistics. Those were introduced by Bousquet-Mélou, Fusy, and Préville-Ratelle who proved they were symmetrically distributed but had no combinatorial explanation. The second bijection sends a Tamari interval to a closed flow of an ordered forest. These combinatorial objects were studied by Chapoton in the context of the Pre-Lie operad and the connection with the Tamari order was still unclear.
The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$
The face numbers of simplicial polytopes that approximate $C^1$-convex bodies in the Hausdorff metric is studied. Several structural results about the skeleta of such polytopes are studied and used to derive a lower bound theorem for this class of polytopes. This partially resolves a conjecture made by Kalai in 1994: if a sequence $\{P_n\}_{n=0}^{\infty}$ of simplicial polytopes converges to a $C^1$-convex body in the Hausdorff distance, then the entries of the $g$-vector of $P_n$ converge to infinity.
We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod{6}$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result.
The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given.
Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients.
We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations.
Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matricesof fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.
Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group $EL_{2n}$ whose positive part $(EL_{2n})_{\geq 0}$ naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical networks. The Lie algebra $el_{2n}$ is semisimple and isomorphic to the symplectic algebra. In the end of their paper, they suggest a generalization of electrical Lie algebras to all finite Dynkin types. We give the structure of the type $B$ electrical Lie algebra $e_{b_{2n}}$. The nonnegative part $(E_{B_{2n}})_{\geq 0}$ of the corresponding Lie group conjecturally acts on a class of "mirror symmetric circular planar electrical networks". This class of networks has interesting combinatorial properties. Finally, we mention some partial results for type $C$ and $D$ electrical Lie algebras, where an analogous story needs to be developed.
The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained.
In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group.
The purpose of this work is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes.
We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in the character formulas for certain representations of the symmetric group and these formulas are usually proven algebraically. Here, we give a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.
We introduce a series of conjectured identities that deform Weyl's denominator formula and generalize Tokuyama's formula to other root systems. These conjectures generalize a number of well-known results due to Okada. We also prove a related result for $B'_n$ that generalizes a theorem of Simpson.
We study an extension of the chip-firing game. A given set of admissible moves, called Yamanouchi moves, allows the player to pass from a starting configuration $\alpha$ to a further configuration $\beta$. This can be encoded via an action of a certain group, the toppling group, associated with each connected graph. This action gives rise to a generalization of Hall-Littlewood symmetric polynomials and a new combinatorial basis for them. Moreover, it provides a general method to construct all orthogonal systems associated with a given random variable.
We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces.
We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators.
Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its <i>arithmetic</i> Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems $A_n, B_n, C_n$, and $D_n$ with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a \emphfinite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.
A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.
We introduce a new approach to the enumeration of rational slope parking functions with respect to the <mathrm>area</mathrm> and a generalized <mathrm>dinv</mathrm> statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (<mathrm>area</mathrm>,<mathrm>dinv</mathrm>) and (<mathrm>bounce</mathrm>, <mathrm>area</mathrm>) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions.
Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.
The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'-$vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'-$vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In this paper we show that these necessary conditions are in fact sufficient to characterize the h'-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.
Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erdős-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect […]