%I
%S 1,1,3,13,79,620,6036,70963
%N The number of KKnuth classes of initial tableaux on n letters.
%C KKnuth equivalence on words is the Ktheoretic 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 Ktheoretic version, two words are said to be KKnuth equivalent if one can be obtained from the other via a finite series of applications of the KKnuth 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 Ktheoretic analog of the wellknown 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 KKnuth equivalence is that, while insertion equivalence via the Schensted algorithm (resp. the Hecke algorithm) implies Knuth equivalence (resp. KKnuth equivalence), the converse holds for the standard version but not for the Ktheoretic version. In other words, two words can be KKnuth 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">KTheory of Minuscule Varieties</a>, arXiv:1306.5419 [math.AG], 2003.
%H Christian Gaetz et al. <a href="http://arxiv.org/abs/1409.6659">KKnuth Equivalence for Increasing Tableaux</a>, preprint arXiv:1409.6659 2015.
%H R. Patrias and P. Pylyavskyy, <a href="http://arxiv.org/abs/1404.4340">KTheoretic PoirerReutenauer Bialgebra</a>, arXiv:1409.6659 [math.CO], 2014.
%H Ka Yu Tam, <a href="/A261601/a261601.py.txt">This program generates KKnuth 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 KTheoretic Schubert Calculus</a>, Algebra Number Theory 3 (2009), no. 2, 121148.
%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 KTheoretic Schubert Calculus</a>, arXiv:0705.2915 [math.CO], 2007.
%e For n = 2, there are 3 KKnuth 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.)
%K nonn,more
%O 0,3
%A _Michelle Mastrianni_, Aug 25 2015
