In mathematics, the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} (depending on two integer partitions λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } . They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Kostka (1882)). For example, if λ = ( 3 , 2 ) {\displaystyle \lambda =(3,2)} and μ = ( 1 , 1 , 2 , 1 ) {\displaystyle \mu =(1,1,2,1)} , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} counts the number of ways to fill a left-aligned collection of boxes with 3 boxes in the first row and 2 boxes 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 K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = 3 {\displaystyle K_{(3,2)(1,1,2,1)}=3} .
Examples and special cases For any partition λ {\displaystyle \lambda } , the Kostka number K λ λ {\displaystyle K_{\lambda \lambda }} is equal to 1: the unique way to fill the Young diagram of shape λ = ( λ 1 , … , λ m ) {\displaystyle \lambda =(\lambda _{1},\dotsc ,\lambda _{m})} with λ 1 {\displaystyle \lambda _{1}} copies of 1, λ 2 {\displaystyle \lambda _{2}} copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape λ {\displaystyle \lambda } .) The Kostka number K λ μ {\displaystyle K_{\lambda \mu }} is positive (i.e., there exist semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } ) if and only if λ {\displaystyle \lambda } and μ {\displaystyle \mu } are both partitions of the same integer n {\displaystyle n} and λ {\displaystyle \lambda } is greater than or equal to μ {\displaystyle \mu } in dominance order. In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if μ = ( 1 , 1 , … , 1 ) {\displaystyle \mu =(1,1,\dotsc ,1)} is the partition whose parts are all 1 then a semistandard Young tableau of weight μ {\displaystyle \mu } is a standard Young tableau; the number of standard Young tableaux of a given shape λ {\displaystyle \lambda } is given by the hook-length formula.
Properties An important simple property of Kostka numbers is that K λ μ {\displaystyle K_{\lambda \mu }} does not depend on the order of entries of μ {\displaystyle \mu } . For example, K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = K ( 3 , 2 ) ( 1 , 1 , 1 , 2 ) {\displaystyle K_{(3,2)(1,1,2,1)}=K_{(3,2)(1,1,1,2)}} . This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weights μ {\displaystyle \mu } and μ ′ {\displaystyle \mu ^{\prime }} , where μ {\displaystyle \mu } and μ ′ {\displaystyle \mu ^{\prime }} differ only by swapping two entries.
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