Sergey Kitaev ; Jeffrey Remmel
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Enumerating (2+2)-free posets by the number of minimal elements and other statistics
dmtcs:2812 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
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https://doi.org/10.46298/dmtcs.2812
Enumerating (2+2)-free posets by the number of minimal elements and other statisticsConference paper
Authors: Sergey Kitaev 1; Jeffrey Remmel 2
0000-0003-3324-1647##NULL
Sergey Kitaev;Jeffrey Remmel
1 The Mathematics Institute, Reyjavik University
2 Department of Mathematics [Univ California San Diego]
A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements.
Combinatorial Structures for Permutation Enumeration and Macdonald Polynomials; Funder: National Science Foundation; Code: 0654060
Bibliographic References
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