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For integers , Brietzke (2008) defined the -central coefficients of an infinite lower triangular matrix as , with , and the -central coefficient triangle of as
It is known that the -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array with and , we obtain the generating function of its -central coefficients and give an explicit representation for the -central Riordan array in terms of the Riordan array . Meanwhile, the...
Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then .
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