Related papers: Promotion permutations and the Robinson--Schensted…
Permutation tableaux are combinatorial objects related with permutations and various statistics on them. They appeared in connection with total positivity in Grassmannians, and stationary probabilities in a PASEP model. In particular they…
We consider two partial orders on standard Young tableaux. The first one is induced from the weak right Bruhat order on symmetric group by Robinson-Schensted algorithm. The second one is induced from the order on Young diagrams by…
We study four bijections, which are promotion, evacuation, rowmotion, and rowvacuation, on generalized Dyck paths in rational Catalan combinatorics. We define the maps on generalized Dyck paths, which have their origins in maps on Dyck…
Brualdi and Ma found a connection between involutions of length $n$ with $k$ descents and symmetric $k\times k$ matrices with non-negative integer entries summing to $n$ and having no row or column of zeros. From their main theorem they…
The two tableaux assigned by the Robinson--Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard…
We give a new (inductive) proof of the classical Frobenius--Young correspondence between irreducible complex representations of the symmetric group and Young diagrams, using the new approach, suggested in \cite{OV, VO}, to determining this…
We establish bijections between three classes of combinatorial objects that have been studied in very different contexts: lattice walks in simplicial regions as introduced by Mortimer--Prellberg, standard cylindric tableaux as introduced by…
The notion of an Abstract Young (briefly: AY) representation is a natural generalization of the classical Young orthogonal form. The AY representations of the symmetric group are characterized by Adin, Brenti and Roichman in [U2]. In this…
We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf…
The hook length formula for $d$-complete posets expresses the number of linear extensions of a $d$-complete poset $P$ in terms of hooks of $P$. It generalizes the usual hook length formula for standard Young tableaux, as well as hook length…
The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin…
Lascoux polynomials are $K$-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux ($\mathsf{RSVT}$) rule for Lascoux polynomials and reverse semistandard Young…
Egecioglu and Remmel gave an interpretation for the entries of the inverse Kostka matrix K^{-1} in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK^{-1}=I but were unable to…
An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a…
The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first…
Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern…
We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple…
This thesis deals with three different aspects of the combinatorics of permutations. In the first two papers, two flavours of pattern avoiding permutations are examined; and in the third paper Young tableaux, which are closely related to…
We explain how to define the Robinson-Schensted-Knuth (RSK) correspondence in terms of local transformations called "toggles." (This note, which is not intended for publication and which is based on presentations of Alex Postnikov, was…
Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to…