We exhibit a weight-preserving bijection between semi-standard Young tableaux and skyline augmented fillings to provide the first combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-SchenstedKnuth Algorithm for skyline augmented fillings. We also prove that ...

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 bijection between rectangular 2-row increasing tableaux and small Schröder paths. Using the jeu de taquin for increasing tableaux of [Thomas–Yong ’09], we 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.

In this paper, we consider the Robinson-Schensted correspondence for oscillating tab-leaux and skew oscillating tableaux deened in 15] and 3]. First we give an analogue, for the oscillating tableaux, of the classical geometric construction of Viennot for standard tableaux ((16]). Then, we extend a construction of Sagan and Stanley ((10]), dealing with standard tableaux and skew tableaux, to ded...

We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-SchenstedKnuth Algorithm for semi-skyline augmented fillings. This procedure co...

We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-SchenstedKnuth Algorithm for semi-skyline augmented fillings. This procedure co...

A tableau inversion is a pair of entries from the same column of a row-standard tableau that lack the relative ordering necessary to make the tableau columnstandard. An i-inverted Young tableau is a row-standard tableau with precisely i inversion pairs, and may be interpreted as a generalization of (column-standard) Young tableaux. Inverted Young tableaux that lack repeated entries were introdu...

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...