Kostka Numbers 2

By a partition of m, we mean an n-tuple: p = [a₁, a₂, .., a_n] of positive integers with a₁ >= a₂ >= .. >= a_n and a₁ + a₂ + .. + a_n = m So to any partion p of m, a Ferrer's diagram consisting of m boxes is associated, arranged in n rows in such a way that i_th row contains a_i boxes. Let c be another partition of m with k-tuple: c = [b₁, b₂, .., b_k] of positive integers with b₁ >= b₂ >= .. >= b_k and b₁ + a₂ + ... + b_k = m If p = [a₁, a₂, .., a_n] and c = [b₁, b₂, .., b_k] are two arbitrary partitions of m, then by a semi-standard Young tableau of shape p and content c, we mean any distribution of the numbers 1, 2, ..,m in the boxes of the associated Ferrer’s diagram of p in such a way that 1. every row is non-decreasing; 2. every column is (strictly) increasing; and 3. for any 1 <= i <= m, the multiplicity of i in the distribution is b_i. For example if m=6: p=[3,2,1] and c=[2,2,2] (which means we have 2 of 1's, 2 of 2's and 2 of 3's, totally 6 of them). 1 1 2 2 3 3 1 1 3 2 2 3 The number of all semi-standard Young tableaux of shape p and content c is denoted by K(p,c) here and it is called the Kostka coefficient or the Kostka number. Let p = [21,13,8,5,3] Find the following: K(p, c₁) where c₁ = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] K(p, c₂) where c₂ = [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] K(p, c₃) where c₃ = [3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2] K(p, c₄) where c₄ = [4,4,4,4,4,4,4,4,4,4,4,4,2] K(p, c₅) where c₅ = [5,5,5,5,5,5,5,5,5,5] Answer format: K(p,c₁),K(p,c₂),K(p,c₃),K(p,c₄),K(p,c₅)