Generalization of Universal Partition and Bipartition Theorems
نویسندگان
چکیده
Let A = (ai,j) , i = 1, 2, . . . , j = 0, 1, 2, . . . , be an infinite matrix with elements ai,j = 0 or 1; p (n, k;A) the number of partitions of n into k parts whose number yi of parts which are equal to i belongs to the set Yi = {j : ai,j = 1} , i = 1, 2, . . . . The universal theorem on partitions states that 1 X n=0 n X k=0 p (n, k;A)ut = 1 Y i=1 0 @ 1 X j=0 ai,ju t 1 A . In this paper, we present a generalization of this result. We show that this generalization remains true when ai,j are indeterminate. We also take into account the bi-partite and multi-partite situations.
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