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A261601
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The number of K-Knuth classes of initial tableaux on n letters.
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1
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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