Abstract
| Language | English |
|---|---|
| Pages | 139-163 |
| Number of pages | 25 |
| Journal | Journal of Combinatorics |
| Volume | 2 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2011 |
Fingerprint
Keywords
- enumerating
- (2+2)-free posets
- indistinguishable elements
Cite this
}
Enumerating (2+2)-free posets by indistinguishable elements. / Dukes, Mark; Kitaev, Sergey; Remmel, Jeffrey; Steingrimsson, Einar.
In: Journal of Combinatorics, Vol. 2, No. 1, 2011, p. 139-163.Research output: Contribution to journal › Article
TY - JOUR
T1 - Enumerating (2+2)-free posets by indistinguishable elements
AU - Dukes, Mark
AU - Kitaev, Sergey
AU - Remmel, Jeffrey
AU - Steingrimsson, Einar
PY - 2011
Y1 - 2011
N2 - 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. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent sequence corresponding to the poset. We derive the generating function for the number of (2+2)-free posets with respect to both maxindist and the number of different strict down-sets of elements in the poset. Moreover, we show that (2+2)-free posets P with maxindist(P) at most k are in bijection with upper triangular matrices of nonnegative integers not exceeding k, where each row and each column contains a nonzero entry. (Here we consider isomorphic posets to be equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1 correspond to upper triangular binary matrices where each row and column contains a nonzero entry, and whose entries sum to n. We derive a generating function counting such matrices, which confirms a conjecture of Jovovic, and we refine the generating function to count upper triangular matrices consisting of nonnegative integers not exceeding k and having a nonzero entry in each row and column. That refined generating function also enumerates (2+2)-free posets according to maxindist. Finally, we link our enumerative results to certain restricted permutations and matrices.
AB - 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. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent sequence corresponding to the poset. We derive the generating function for the number of (2+2)-free posets with respect to both maxindist and the number of different strict down-sets of elements in the poset. Moreover, we show that (2+2)-free posets P with maxindist(P) at most k are in bijection with upper triangular matrices of nonnegative integers not exceeding k, where each row and each column contains a nonzero entry. (Here we consider isomorphic posets to be equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1 correspond to upper triangular binary matrices where each row and column contains a nonzero entry, and whose entries sum to n. We derive a generating function counting such matrices, which confirms a conjecture of Jovovic, and we refine the generating function to count upper triangular matrices consisting of nonnegative integers not exceeding k and having a nonzero entry in each row and column. That refined generating function also enumerates (2+2)-free posets according to maxindist. Finally, we link our enumerative results to certain restricted permutations and matrices.
KW - enumerating
KW - (2+2)-free posets
KW - indistinguishable elements
UR - http://intlpress.com/site/pub/pages/journals/items/joc/content/vols/0002/0001/a006/index.html
UR - http://arxiv.org/abs/1006.2696
U2 - 10.4310/JOC.2011.v2.n1.a6
DO - 10.4310/JOC.2011.v2.n1.a6
M3 - Article
VL - 2
SP - 139
EP - 163
JO - Journal of Combinatorics
T2 - Journal of Combinatorics
JF - Journal of Combinatorics
SN - 2156-3527
IS - 1
ER -