Some remarks on sign-balanced and maj-balanced posets

Details Some remarks on sign-balanced and maj-balanced posets Some remarks on sign-balanced and maj-balanced posets

2002-11-06

arxiv.org/abs/math/0211113v6

Mathematics

Combinatorics

30 pages. Some inaccuracies in Section 3 have been corrected, and Conjecture 3.6 has been added

Scientific paper

Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the work of Frank Ruskey, who was interested in the efficient generation of all linear extensions of P. We survey a number of techniques for showing that posets are sign-balanced, and more generally, computing their "imbalance." There are close connections with domino tilings and, for certain posets, a "domino generalization" of Schur functions due to Carre and Leclerc. We also say that P is maj-balanced if exactly half the linear extensions of P have even major index. We discuss some similarities and some differences between sign-balanced and maj-balanced posets.

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