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Bender–Knuth involution
In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
==Definition==
The Bender–Knuth involutions σ''k'' are defined for integers ''k'', and act on the set of semistandard skew Young tableaux of some fixed shape μ/ν, where μ and ν are partitions. It acts by changing some of the elements ''k'' of the tableau to ''k'' + 1, and some of the entries ''k'' + 1 to ''k'', in such a way that the numbers of elements with values ''k'' or ''k'' + 1 are exchanged. Call an entry of the tableau free if it is ''k'' or ''k'' + 1 and there is no other element with value ''k'' or ''k'' + 1 in the same column. For any ''i'', the free entries of row ''i'' are all in consecutive columns, and consist of ''a''''i'' copies of ''k'' followed by ''b''''i'' copies of ''k'' + 1, for some ''a''''i'' and ''b''''i''. The Bender–Knuth involution σ''k'' replaces them by
''b''''i'' copies of ''k'' followed by ''a''''i'' copies of ''k'' + 1.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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