In the study of lattice walks there are several examples of enumerative equivalences which amount to a trade-o between domain and endpoint constraints. We present a family of such bijections for simple walks in Weyl chambers which use arc diagrams in a natural way. One consequence is a set of new bijections for standard Young tableaux of bounded height. A modi cation of the argument in two dime...

The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functio...

Abstract. We present some conjectures and open problems on partition hook lengths, which are all motivated by known results on the subject. The conjectures are suggested by extensive experimental calculations using a computer algebra system. The first conjecture unifies two classical results on the number of standard Young tableaux and the number of pairs of standard Young tableaux of the same ...

The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functio...

We study Gröbner degenerations of Schubert varieties inside flag varieties. consider toric induced by matching fields and semi-standard Young tableaux. describe an analogue field ideals for the variety give a complete characterization among them. use combinatorial approach to standard monomial theory show that block diagonal rise degenerations. Our methods results combinatorics permutations ass...

Oscillating tableaux are certain walks in Young’s lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all t...

The number of words w = w1 · · ·wn, 1 ≤ wi ≤ k, for which there are 1 ≤ i1 < · · · < i` ≤ n and wi1 > · · · > wi` , is given, by the Schensted-Knuth correspondence, in terms of standard and semi-standard Young tableaux. When n → ∞, the asymptotics of the number of such words is calculated. Work partially supported by N.S.F. Grant No. DMS-94-01197.

Stanley conjectured that the number of maximal chains in the weak Bruhat order of Sn, or equivalently the number of reduced decompositions of the reverse of the identity permutation w0 = n, n− 1, n − 2, . . . , 2, 1, equals the number of standard Young tableaux of staircase shape s = {n− 1, n− 2, . . . , 1}. Originating from this conjecture remarkable connections between standard Young tableaux...

A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equidistributed with the standard one. This concept then applied construct explicit cyclic extensions Young tableaux Motzkin paths. Schur-positivity of associated quasisymmetric functions follows.