%N The number of K-Knuth classes of initial tableaux on n letters.
%C 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:
%C xzy ~ zxy, (x < y < z)
%C yxz ~ yzx, (x < y < z).
%C 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:
%C xzy ~ zxy, (x < y < z)
%C yxz ~ yzx, (x < y < z)
%C x ~ xx,
%C xyx ~ yxy.
%C 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.
%H A. Buch and M. Samuel, <a href="http://arxiv.org/abs/1306.5419">K-Theory of Minuscule Varieties</a>, arXiv:1306.5419 [math.AG], 2003.
%H Christian Gaetz et al. <a href="http://arxiv.org/abs/1409.6659">K-Knuth Equivalence for Increasing Tableaux</a>, preprint arXiv:1409.6659 2015.
%H R. Patrias and P. Pylyavskyy, <a href="http://arxiv.org/abs/1404.4340">K-Theoretic Poirer-Reutenauer Bialgebra</a>, arXiv:1409.6659 [math.CO], 2014.
%H Ka Yu Tam, <a href="/A261601/a261601.py.txt">This program generates K-Knuth equivalence data for initial tableaux on [n].</a>
%H H. Thomas and A. Yong, <a href="http://dx.doi.org/10.2140/ant.2009.3.121">A Jeu de Taquin Theory for Increasing Tableaux, with Applications to K-Theoretic Schubert Calculus</a>, Algebra Number Theory 3 (2009), no. 2, 121-148.
%H H. Thomas and A. Yong, <a href="http://arxiv.org/abs/0705.2915">A Jeu de Taquin Theory for Increasing Tableaux, with Applications to K-Theoretic Schubert Calculus</a>, arXiv:0705.2915 [math.CO], 2007.
%e 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).
%e (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.)
%A _Michelle Mastrianni_, Aug 25 2015