Some remarks on sign-balanced and maj-balanced posets

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

Partition identity bijections related to sign-balance and rank

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 81-83).In this thesis, we present bijections proving partitions identities. In the first part, we generalize Dyson's definition of rank to partitions with successive Durfee squares. We then present two symmetries for this new rank which we prove using bijections generalizing conjugation and Dyson's map. Using these two symmetries we derive a version of Schur's identity for partitions with successive Durfee squares and Andrews' generalization of the Rogers-Ramanujan identities. This gives a new combinatorial proof of the first Rogers-Ramanujan identity. We also relate this work to Garvan's generalization of rank. In the second part, we prove a family of four-parameter partition identities which generalize Andrews' product formula for the generating function for partitions with respect number of odd parts and number of odd parts of the conjugate. The parameters which we use are related to Stanley's work on the sign-balance of a partition.by Cilanne Emily Boulet.Ph.D