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In mathematics, the Kostka number ''K''λμ (depending on two integer partitions λ and μ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They were introduced by the mathematician Carl Kostka in his study of symmetric functions ().〔Stanley, Enumerative combinatorics, volume 2, p. 398.〕 For example, if λ = (3, 2) and μ = (1, 1, 2, 1), the Kostka number ''K''λμ 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 ''K''(3, 2) (1, 1, 2, 1) = 3. ==Examples and special cases== For any partition λ, the Kostka number ''K''λλ is equal to 1: the unique way to fill the Young diagram of shape λ = (λ1, λ2, ..., λ''m'') with λ1 copies of 1, λ2 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.) The Kostka number ''K''λμ is positive (i.e., there exist semistandard Young tableaux of shape λ and weight μ) if and only if λ and μ are both partitions of the same integer ''n'' and λ is larger than μ in dominance order.〔Stanley, Enumerative combinatorics, volume 2, p. 315.〕 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, ..., 1) is the partition whose parts are all 1 then a semistandard Young tableau of weight μ is a standard Young tableau; the number of standard Young tableaux of a given shape λ is given by the hook-length formula. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kostka number」の詳細全文を読む スポンサード リンク
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