A261601 The number of K-Knuth classes of initial tableaux on n letters.

1, 1, 3, 13, 79, 620, 6036, 70963

Offset

0,3

Comments

K-Knuth equivalence on words is the K-theoretic analog for Knuth equivalence on words. Two words are said to be Knuth equivalent if one can be obtained from the other via a finite series of applications of the Knuth relations:

xzy ~ zxy, (x < y < z)

yxz ~ yzx, (x < y < z).

In the K-theoretic version, two words are said to be K-Knuth equivalent if one can be obtained from the other via a finite series of applications of the K-Knuth relations:

xzy ~ zxy, (x < y < z)

yxz ~ yzx, (x < y < z)

x ~ xx,

xyx ~ yxy.

In 2006, Buch et al. introduced a new combinatorial algorithm called Hecke insertion, which is a K-theoretic analog of the well-known Schensted algorithm for the insertion of a word into a semistandard Young tableau. The Hecke insertion algorithm results in a strictly increasing tableau. An important difference between Knuth equivalence and K-Knuth equivalence is that, while insertion equivalence via the Schensted algorithm (resp. the Hecke algorithm) implies Knuth equivalence (resp. K-Knuth equivalence), the converse holds for the standard version but not for the K-theoretic version. In other words, two words can be K-Knuth equivalent but insert into different tableaux via the Hecke insertion algorithm.

Links

Table of n, a(n) for n=0..7.

A. Buch and M. Samuel, K-Theory of Minuscule Varieties, arXiv:1306.5419 [math.AG], 2003.

Christian Gaetz et al. K-Knuth Equivalence for Increasing Tableaux, preprint arXiv:1409.6659 2015.

R. Patrias and P. Pylyavskyy, K-Theoretic Poirer-Reutenauer Bialgebra, arXiv:1409.6659 [math.CO], 2014.

Ka Yu Tam, This program generates K-Knuth equivalence data for initial tableaux on [n].

H. Thomas and A. Yong, A Jeu de Taquin Theory for Increasing Tableaux, with Applications to K-Theoretic Schubert Calculus, Algebra Number Theory 3 (2009), no. 2, 121-148.

H. Thomas and A. Yong, A Jeu de Taquin Theory for Increasing Tableaux, with Applications to K-Theoretic Schubert Calculus, arXiv:0705.2915 [math.CO], 2007.

Example

For n = 2, there are 3 K-Knuth classes, each with one tableau. The tableaux representing the classes are the minimal tableaux of partition shapes (2), (1,1), and (2,1).
(A minimal tableau is a tableau in which each box is filled with the smallest positive integer that will make the filling a valid strictly increasing tableau.)

Crossrefs

Sequence in context: A159312 A355165 A213527 * A125659 A010844 A090364

Adjacent sequences: A261598 A261599 A261600 * A261602 A261603 A261604

Keyword

nonn,more

Author

Michelle Mastrianni, Aug 25 2015

Status

approved