34 research outputs found
A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1
whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We
present a unifying perspective on ASMs and other combinatorial objects by
studying a certain tetrahedral poset and its subposets. We prove the order
ideals of these subposets are in bijection with a variety of interesting
combinatorial objects, including ASMs, totally symmetric self-complementary
plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux,
Catalan objects, tournaments, and totally symmetric plane partitions. We prove
product formulas counting these order ideals and give the rank generating
function of some of the corresponding lattices of order ideals. We also prove
an expansion of the tournament generating function as a sum over TSSCPPs. This
result is analogous to a result of Robbins and Rumsey expanding the tournament
generating function as a sum over alternating sign matrices.Comment: 24 pages, 23 figures, full published version of arXiv:0905.449
The toggle group, homomesy, and the Razumov-Stroganov correspondence
The Razumov-Stroganov correspondence, an important link between statistical
physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello,
relates the ground state eigenvector of the O(1) dense loop model on a
semi-infinite cylinder to a refined enumeration of fully-packed loops, which
are in bijection with alternating sign matrices. This paper reformulates a key
component of this proof in terms of posets, the toggle group, and homomesy, and
proves two new homomesy results on general posets which we hope will have
broader implications.Comment: 14 pages, 10 figures, final versio
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
A direct bijection between descending plane partitions with no special parts and permutation matrices
We present a direct bijection between descending plane partitions with no
special parts and permutation matrices. This bijection has the desirable
property that the number of parts of the descending plane partition corresponds
to the inversion number of the permutation. Additionally, the number of maximum
parts in the descending plane partition corresponds to the position of the one
in the last column of the permutation matrix. We also discuss the possible
extension of this approach to finding a bijection between descending plane
partitions and alternating sign matrices.Comment: 8 pages, 4 figures. Final versio