: Mathématiques: Kostka number

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Kostka number

In mathematics, the Kostka number ${\\displaystyle K_{\\lambda \\mu }}$ (depending on two integer partitions ${\\displaystyle \\lambda }$ and ${\\displaystyle \\mu }$ ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape ${\\displaystyle \\lambda }$ and weight ${\\displaystyle \\mu }$ . They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Kostka (1882)). For example, if ${\\displaystyle \\lambda =(3,2)}$ and ${\\displaystyle \\mu =(1,1,2,1)}$ , the Kostka number ${\\displaystyle K_{\\lambda \\mu }}$ counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and ${\\displaystyle K_{(3,2)(1,1,2,1)}=3}$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Kostka_number)