In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by Bender & Knuth (1972, pp.
46–47) in their study of plane partitions.
The Bender–Knuth involutions
are defined for integers
, and act on the set of semistandard skew Young tableaux of some fixed shape
μ
ν
μ
It acts by changing some of the elements
of the tableau to
, in such a way that the numbers of elements with values
are exchanged.
Call an entry of the tableau free if it is
and there is no other element with value
, the free entries of row
are all in consecutive columns, and consist of
The Bender–Knuth involution
Bender–Knuth involutions can be used to show that the number of semistandard skew tableaux of given shape and weight is unchanged under permutations of the weight.
In turn this implies that the Schur function of a partition is a symmetric function.
Bender–Knuth involutions were used by Stembridge (2002) to give a short proof of the Littlewood–Richardson rule.