The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer. For which multisubsets \(B\) of the symmetric group \(S_n\) is the quasisymmetric function \(Q(B) = \sum_{\pi \in B}F_{Des(\pi), n}\) a symmetric function? Here \(Des(\pi)\) is…
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On cyclic Schur-positive sets of permutations
Published: w/ S. Elizalde, Y. Roichman, Electron. J. Combin, 2020.
Revisiting pattern avoidance and quasisymmetric functions
Let \(S_n\) be the nth symmetric group. Given a set of permutations \(\Pi\) we denote by \(S_n(\Pi)\) the set of permutations in \(S_n\) which avoid \(\Pi\) in the sense of pattern avoidance. Consider the generating function \(Q_n(\Pi) = \sum_{\pi} F_{Des…
Counting pattern-avoiding integer partitions
A partition \(\alpha\) is said to contain another partition (or pattern) \(\mu\) if the Ferrers board for \(\mu\) is attainable from \(\alpha\) under removal of rows and columns. We say \(\alpha\) avoids \(\mu\) if it does not contain \(\mu\). In…
On criteria for rook equivalence of Ferrers boards
In [2] we introduced a new notion of Wilf equivalence of integer partitions and proved that rook equivalence implies Wilf equivalence. In the present paper we prove the converse and thereby establish a new criterion for rook equivalence. We also…
Rook and Wilf equivalence of integer partitions
The subjects of rook equivalence and Wilf equivalence have both attracted considerable attention over the last half-century. In this paper we introduce a new notion of Wilf equivalence for integer partitions, and, using this notion, we prove that rook equivalence…