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 A261601 The number of K-Knuth classes of initial tableaux on n letters. 1
 1, 1, 3, 13, 79, 620, 6036, 70963 (list; graph; refs; listen; history; text; internal format)
 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 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. 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: A215915 A159312 A213527 * A125659 A010844 A090364 Adjacent sequences:  A261598 A261599 A261600 * A261602 A261603 A261604 KEYWORD nonn,more AUTHOR Michelle Mastrianni, Aug 25 2015 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)