DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)

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1. Partitioned Cacti: a Bijective Approach to the Cycle Factorization Problem

Gilles Schaeffer ; Ekaterina Vassilieva.

In this paper we construct a bijection for partitioned 3-cacti that gives raise to a new formula for enumeration of factorizations of the long cycle into three permutations with given number of cycles.

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2. A Quantization of a theorem of Goulden and Jackson

Matjaž Konvalinka ; Mark Skandera.

A theorem of Goulden and Jackson which gives interesting formulae for character immanants also implies MacMahon's Master Theorem. We quantize Goulden and Jackson's theorem to give formulae for quantum character immanants in such a way as to obtain a known quantization of MacMahon's Master Theorem due to Garoufalidis-Lê-Zeilberger. In doing so, we also quantize formulae of Littlewood, Merris and Watkins concerning induced character immanants.

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3. Affine descents and the Steinberg torus

Kevin Dilks ; T. Kyle Petersen ; John R. Stembridge.

Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.

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4. Pattern-Avoidance in Binary Fillings of Grid Shapes (short version)

Alexey Spiridonov.

A $\textit{grid shape}$ is a set of boxes chosen from a square grid; any Young diagram is an example. This paper considers a notion of pattern-avoidance for $0-1$ fillings of grid shapes, which generalizes permutation pattern-avoidance. A filling avoids some patterns if none of its sub-shapes equal any of the patterns. We focus on patterns that are $\textit{pairs}$ of $2 \times 2$ fillings. For some shapes, fillings that avoid specific $2 \times 2$ pairs are in bijection with totally nonnegative Grassmann cells, or with acyclic orientations of bipartite graphs. We prove a number of results analogous to Wilf-equivalence for these objects ―- that is, we show that for certain classes of shapes, some pattern-avoiding fillings are equinumerous with others.

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5. $n$-color overpartitions, lattice paths, and multiple basic hypergeometric series

Olivier Mallet.

We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions.

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6. Algebraic shifting and strongly edge decomposable complexes

Satoshi Murai.

Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

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7. Skew domino Schensted algorithm and sign-imbalance

Jang Soo Kim.

Using growth diagrams, we define a skew domino Schensted algorithm which is a domino analogue of the "Robinson-Schensted algorithm for skew tableaux'' due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for a weighted sum of skew domino tableaux whose special case is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula.

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8. Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees

Itamar Landau ; Lionel Levine ; Yuval Peres.

The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.

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9. Cyclic Sieving, Promotion, and Representation Theory

Brendon Rhoades.

We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions.

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10. New combinatorial computational methods arising from pseudo-singletons

Gilbert Labelle.

Since singletons are the connected sets, the species $X$ of singletons can be considered as the combinatorial logarithm of the species $E(X)$ of finite sets. In a previous work, we introduced the (rational) species $\widehat{X}$ of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that $E(X) = \exp (\widehat{X})$ in the context of rational species, where $\exp (T)$ denotes the classical analytical power series for the exponential function in the variable $T$. In the present work, we use the species $\widehat{X}$ to create new efficient recursive schemes for the computation of molecular expansions of species of rooted trees, of species of assemblies of structures, of the combinatorial logarithm species, of species of connected structures, and of species of structures with weighted connected components.

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11. $(\ell, 0)$-Carter Partitions and their crystal theoretic interpretation

Chris Berg ; Monica Vazirani.

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). The condition of being an $(\ell,0)$-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding Specht module over the finite Hecke algebra. We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph of $B(\Lambda_0)$.

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12. A bijection between shrubs and series-parallel posets

Frédéric Chapoton.

Motivated by the theory of operads, we introduce new combinatorial objects, called shrubs, that generalize forests of rooted trees. We show that the species of shrubs is isomorphic to the species of series-parallel posets.

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13. From Bruhat intervals to intersection lattices and a conjecture of Postnikov

Axel Hultman ; Svante Linusson ; John Shareshian ; Jonas Sjöstrand.

We prove the conjecture of A. Postnikov that ($\mathrm{A}$) the number of regions in the inversion hyperplane arrangement associated with a permutation $w \in \mathfrak{S}_n$ is at most the number of elements below $w$ in the Bruhat order, and ($\mathrm{B}$) that equality holds if and only if $w$ avoids the patterns $4231$, $35142$, $42513$ and $351624$. Furthermore, assertion ($\mathrm{A}$) is extended to all finite reflection groups.

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14. Enumeration of orientable coverings of a non-orientable manifold

Jin Ho Kwak ; Alexander Mednykh ; Roman Nedela.

In this paper we solve the known V.A. Liskovets problem (1996) on the enumeration of orientable coverings over a non-orientable manifold with an arbitrary finitely generated fundamental group. As an application we obtain general formulas for the number of chiral and reflexible coverings over the manifold. As a further application, we count the chiral and reflexible maps and hypermaps on a closed orientable surface by the number of edges. Also, by this method the number of self-dual and Petri-dual maps can be determined. This will be done in forthcoming papers by authors.

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15. Graph weights arising from Mayer and Ree-Hoover theories of virial expansions

Amel Kaouche ; Pierre Leroux.

We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families.

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16. Deodhar Elements in Kazhdan-Lusztig Theory

Brant Jones.

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered.

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17. On the link pattern distribution of quarter-turn symmetric FPL configurations

Philippe Duchon.

We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.

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18. Multi-triangulations as complexes of star polygons

Vincent Pilaud ; Francisco Santos.

A $k$-triangulation of a convex polygon is a maximal set of diagonals so that no $k+1$ of them mutually cross. $k$-triangulations have received attention in recent literature, with motivation coming from several interpretations of them. We present a new way of looking at $k$-triangulations, where certain star polygons naturally generalize triangles for $k$-triangulations. With this tool we give new, direct proofs of the fundamental properties of $k$-triangulations (number of edges, definition of flip). This interpretation also opens up new avenues of research that we briefly explore in the last section.

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19. Distances in random Apollonian network structures

Olivier Bodini ; Alexis Darrasse ; Michèle Soria.

In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we prove a Rayleigh limit distribution for distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order $n$ is asymptotically $\sqrt{n}$.

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20. Flag enumerations of matroid base polytopes

Sangwook Kim.

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the $\textbf{cd}$-index of a polytope can be expressed when a polytope is cut by a hyperplane, and apply these to the $\textbf{cd}$-index of a matroid base polytope of a rank $2$ matroid.

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21. $q,t$-Fuß-Catalan numbers for complex reflection groups

Christian Stump.

In type $A$, the $q,t$-Fuß-Catalan numbers $\mathrm{Cat}_n^{(m)}(q,t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $\mathcal{S}_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.

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22. Polyominoes determined by involutions

Filippo Disanto ; Simone Rinaldi.

A permutomino of size n is a polyomino determined by particular pairs $(\pi_1, \pi_2)$ of permutations of length $n$, such that $\pi_1(i) \neq \pi_2(i)$, for $1 \leq i \leq n$. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal $x = y$. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated to these objects as particular involutions of length $n$.

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23. Nested Hilbert Schemes and the nested $q,t$-Catalan series

Mahir Bilen Can.

In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series.

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24. Pattern avoidance in dynamical systems

José María Amigó ; Sergi Elizalde ; Matthew B. Kennel.

Orbits generated by discrete-time dynamical systems have some interesting combinatorial properties. In this paper we address the existence of forbidden order patterns when the dynamics is generated by piecewise monotone maps on one-dimensional closed intervals. This means that the points belonging to a sufficiently long orbit cannot appear in any arbitrary order. The admissible patterns are then (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions. The last part of the paper studies and enumerates forbidden order patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. In spite of their simple structure, shift systems exhibit all important features of low-dimensional chaos, allowing to export the results to other dynamical systems via order-isomorphisms. This paper summarizes some results from [1] and [11].

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25. Equivalences for pattern avoiding involutions and classification

Mark Dukes ; Vít Jelínek ; Toufik Mansour ; Astrid Reifegerste.

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions.

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26. Combinatorial Hopf Algebras and Towers of Algebras

Nantel Bergeron ; Thomas Lam ; Huilan Li.

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n \geq 0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n \geq 0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim (A_n)=r^nn!$ where $r = \dim (A_1)$.

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27. Total positivity for cominuscule Grassmannians

Thomas Lam ; Lauren Williams.

In this paper we explore the combinatorics of the non-negative part $(G/P)_{\geq 0}$ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams ― certain fillings of generalized Young diagrams which are in bijection with the cells of $(G/P)_{\geq 0}$. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively.

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28. Bijections for Permutation Tableaux

Sylvie Corteel ; Philippe Nadeau.

In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6].

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29. The Mukhin―Varchenko conjecture for type $A$

S. Ole Warnaar.

We present a generalisation of the famous Selberg integral. This confirms the $\mathfrak{g}=A_n$ case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra $\mathfrak{g}$.

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30. Combinatorial interpretation and positivity of Kerov's character polynomials

Valentin Féray.

Kerov's polynomials give irreducible character values of the symmetric group in term of the free cumulants of the associated Young diagram. Using a combinatorial approach with maps, we prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov.

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31. New bijective links on planar maps

Eric Fusy.

This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections $\Phi _{1,2}$ are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, $\Phi_{1,2}$ yield two new bijections $F_{1,2}$ between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, $\Phi_2 \circ \Phi_1$ specialises to a bijection $F_1$ between 2-connected maps and irreducible triangulations; (ii) $F_1$ gives rise to a bijection $F_2$ between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way.

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32. Families of prudent self-avoiding walks

Mireille Bousquet-Mélou.

A self-avoiding walk on the square lattice is $\textit{prudent}$, if it never takes a step towards a vertex it has already visited. Préa was the first to address the enumeration of these walks, in 1997. For 4 natural classes of prudent walks, he wrote a system of recurrence relations, involving the length of the walks and some additional "catalytic'' parameters. The generating function of the first class is easily seen to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (FPSAC'05). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even $D$-finite. The fourth class ―- general prudent walks ―- still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-$D$-finite. We also study the end-to-end distance of these walks and provide random generation procedures.

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33. A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

S. Assaf.

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.

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34. Affine and toric arrangements

Richard Ehrenborg ; Margaret Readdy ; Michael Slone.

We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions.

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35. Enumerating alternating tree families

Markus Kuba ; Alois Panholzer.

We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results.

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36. The complexity of computing Kronecker coefficients

Peter Bürgisser ; Christian Ikenmeyer.

Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group $S_n$. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem $\mathrm{KRONCOEFF}$ of computing Kronecker coefficients is very difficult. More specifically, we prove that $\mathrm{KRONCOEFF}$ is #$\mathrm{P}$-hard and contained in the complexity class $\mathrm{GapP}$. Formally, this means that the existence of a polynomial time algorithm for $\mathrm{KRONCOEFF}$ is equivalent to the existence of a polynomial time algorithm for evaluating permanents.

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37. Hecke group algebras as degenerate affine Hecke algebras

Florent Hivert ; Anne Schilling ; Nicolas M. Thiéry.

The Hecke group algebra $\operatorname{H} \mathring{W}$ of a finite Coxeter group $\mathring{W}$, as introduced by the first and last author, is obtained from $\mathring{W}$ by gluing appropriately its $0$-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $\mathring{W}$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $\operatorname{H} \mathring{W}$ is the natural quotient of the affine Hecke algebra $\operatorname{H}(W)(q)$ through its level $0$ representation. The proof relies on the following core combinatorial result: at level $0$ the $0$-Hecke algebra acts transitively on $\mathring{W}$. Equivalently, in type $A$, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level $0$ representation is a calibrated principal series representation $M(t)$ for a suitable choice of character $t$, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical $0$-Hecke algebra and that of the affine Hecke algebra at this specialization.

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38. The quasiinvariants of the symmetric group

Jason Bandlow ; Gregg Musiker.

For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$.

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39. The Rees product of the cubical lattice with the chain

Patricia Muldoon ; Margaret A. Readdy.

We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof.

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40. On Plücker coordinates of a perfectly oriented planar network

Kelli Talaska.

Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.

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41. Combinatorial properties of permutation tableaux

Alexander Burstein ; Niklas Eriksen.

We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between $1$-hinge and $0$-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that have previously been shown to be equidistributed by mapping them to patterns in related permutations. We give two direct maps on tableaux that prove the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux.

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42. Enumeration of bilaterally symmetric 3-noncrossing partitions

Guoce Xin ; Terence Y. J. Zhang.

Schützenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for $3$-noncrossing partitions, we use a different technique to develop a $\mathsf{MAPLE}$ package for $2$-dimensional vacillating lattice walk enumeration problems. As an application, we find an interesting relation between two special bilaterally symmetric partitions.

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43. Combinatorial Gelfand Models

Ron M. Adin ; Alex Postnikov ; Yuval Roichman.

A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.

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44. Nestings of Matchings and Permutations and North Steps in PDSAWs

Martin Rubey.

We present a simple bijective proof of the fact that matchings of $[2n]$ with N nestings are equinumerous to $\textit{partially directed self avoiding walks}$ confined to the symmetric wedge defined by $y= \pm x$, with $n$ east steps and $N$ north steps. A very similar construction connects permutations with $N$ nestings and $\textit{PDSAWs}$ remaining below the $x$-axis, again with $N$ north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.

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45. Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups

Anouk Bergeron-Brlek.

We consider the graded Hopf algebra $NCSym$ of symmetric functions with non-commutative variables, which is analogous to the algebra $Sym$ of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the $Sym$ bases and expressions for a shuffle product on $NCSym$. We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results.

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46. A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

Miguel Méndez ; Adolfo Rodríguez.

We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.

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47. Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

François Bergeron ; Aaron Lauve.

We analyze the structure of the algebra $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. In the case $|\mathbf{x}|= \infty$, our techniques simplify to a form readily generalized to many other familiar pairs of combinatorial Hopf algebras.

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48. Double Schubert polynomials for the classical Lie groups

Takeshi Ikeda ; Leonardo Mihalcea ; Hiroshi Naruse.

For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov.

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49. Quasisymmetric Schur functions

James Haglund ; Sarah Mason ; Kurt Luoto ; Steph van Willigenburg.

We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions.

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50. Perfect Matchings and Cluster Algebras of Classical Type

Gregg Musiker.

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases.

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51. Quivers and the Euclidean algebra (Extended abstract)

Alistair Savage.

We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations.

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52. The Sorting Order on a Coxeter Group

Drew Armstrong.

Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.

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53. Staircase Macdonald polynomials and the $q$-Discriminant

Adrien Boussicault ; Jean-Gabriel Luque.

We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}_k(\mathbb{X};q)$ on different bases of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis.

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54. Delannoy numbers and Legendre polytopes

Gábor Hetyei.

We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par

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55. The cluster basis $\mathbb{Z}[x_{1,1},…,x_{3,3}]

Mark Skandera.

We show that the set of cluster monomials for the cluster algebra of type $D_4$ contains a basis of the $\mathbb{Z}$-module $\mathbb{Z}[x_{1,1},\ldots ,x_{3,3}]$. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element as a product of cluster variables.

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56. A generalization of $(q,t)$-Catalan and nabla operators

N. Bergeron ; F. Descouens ; M. Zabrocki.

We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths.

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57. A variation on the tableau switching and a Pak-Vallejo's conjecture

Olga Azenhas.

Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the Littlewood-Richardson coefficients $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and will be shown to be equivalent to the first one defined by the involution property of the Benkart-Sottile-Stroomer tableau switching. The proof of this equivalence provides, in the case the first tableau is Yamanouchi, a variation of the tableau switching algorithm which shows $\textit{switching}$ as an operation that takes two tableaux sharing a common border and moves them trough each other by decomposing the first tableau into a sequence of tableaux whose sequence of partition shapes defines a Gelfand-Tsetlin pattern. This property leads to a $\textit{jeu de taquin-chain sliding}$ on Littlewood-Richardson tableaux.

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58. Minimal Factorizations of Permutations into Star Transpositions

J. Irving ; A. Rattan.

We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form $(1 i)$. Our result generalizes earlier work of Pak ($\textit{Reduced decompositions of permutations in terms of star transpositions, generalized catalan numbers and k-ary trees}$, Discrete Math. $\textbf{204}$:329―335, 1999) in which substantial restrictions were placed on the permutation being factored.

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59. Classification of bijections between 321- and 132-avoiding permutations

Anders Claesson ; Sergey Kitaev.

It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs confirming this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and we show how they are related to each other (via "trivial'' bijections). Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also analyze how many permutation statistics (from a fixed, but large, set of statistics) each of the known bijections preserves, obtaining substantial extensions of known results. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics (the largest number of statistics any of the bijections respect).

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60. A bijective proof of a factorization formula for Macdonald polynomials at roots of unity

F. Descouens ; H. Morita ; Y. Numata.

We give a combinatorial proof of the factorization formula of modified Macdonald polynomials $\widetilde{H}_{\lambda} (X;q,t)$ when $t$ is specialized at a primitive root of unity. Our proof is restricted to the special case where $\lambda$ is a two columns partition. We mainly use the combinatorial interpretation of Haiman, Haglund and Loehr giving the expansion of $\widetilde{H}_{\lambda} (X;q,t)$ on the monomial basis.

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61. Reduction formulae from the factorization Theorem of Littlewood-Richardson polynomials by King, Tollu and Toumazet

Soojin Cho ; Eun-Kyoung Jung ; Dongho Moon.

The factorization theorem by King, Tollu and Toumazet gives four different reduction formulae of Littlewood-Richardson coefficients. One of them is the classical reduction formula of the first type while others are new. Moreover, the classical reduction formula of the second type is not a special case of KTT theorem. We give combinatorial proofs of reduction formulae in terms of tableaux or hives. The proofs for the cases $r=1, 2, n-2$ in terms of tableaux and the proof for the classical reduction formula of the second type in terms of hives are new.

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