The Robinson-Schensted-Knuth correspondence (RSK, see [8] and Corollary 2.5 below) is a bijection between pairs of semi-standard Young tableaux of the same shape and matrices with nonnegative integer entries with prescribed column and row sums. This correspondence plays an important role in the representation theory of the symmetric group and general linear groups, and in the theory of symmetri...

Another bijective proof of Stanley’s hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author’s previous bijective proof (“An involution principle-free bijective proof of Stanley’s hook-content formula”, Di...

We introduce several analogs of the Robinson-Schensted algorithm for skew Young tableaux. These correspondences provide combinatorial proofs of various identities involving f&,, the number of standard skew tableaux of shape L/p, and the skew Schur functions So..,,. For example, we are able to show bijectively that and 4 S;&)QdY)=C ~p;p(x)~~,p(Y) n (1-%Y,)Y. P 1. I It is then shown that these ne...

We make a systematic study of a new combinatorial construction called a dual equivalence graph. Motivated by the dual equivalence relation on standard Young tableaux introduced by Haiman, we axiomatize such constructions and prove that the generating functions of these graphs are Schur positive. We construct a graph on k-ribbon tableaux which we conjecture to be a dual equivalence graph, and we...

A specialisation of a transformation formula for multi-dimensional elliptic hypergeometric series is used to provide compact, non-determinantal formulae for the generating function with respect to the major index of standard Young tableaux of skew shapes of the form “staircase minus rectangle”.

The new Combinatorica provides functions for enumerating, selecting, ranking, and unranking various combinatorial objects such as permutations, combinations, integer partitions, set partitions, Young tableaux, trees, and graphs. It also provides functions to generate various classes of graphs and provides functions for all the standard graph algorithms. The specific ways in which the new Combin...

We rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of Young tableaux and integer partitions. In this way we provide a proof of a result, previously only conjectured, for the total number of stars.

A chess tableau is a standard Young tableau in which, for all i and j, the parity of the entry in cell (i, j) equals the parity of i + j + 1. Chess tableaux were first defined by Jonas Sjöstrand in his study of the sign-imbalance of certain posets, and were independently rediscovered by the authors less than a year later in the completely different context of composing chess problems with inter...

which has a straightforward combinatorial proof when an is the number of involutions of [n], must also hold when an is the number of Young tableaux on [n]. In this paper, we first give (Section 3) a combinatorial proof of (1) for Young tableaux which agrees under the Robinson-Schensted correspondence with the proof for involutions. The proof involves a recursive construction which depends in pa...