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( m , r ) -central Riordan arrays and their applications

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

For integers m > r ≥ 0 , Brietzke (2008) defined the ( m , r ) -central coefficients of an infinite lower triangular matrix G = ( d , h ) = ( d n , k ) n , k ∈ ℕ as d m n + r , ( m - 1 ) n + r , with n = 0 , 1 , 2 , ⋯ , and the ( m , r ) -central coefficient triangle of G as G ( m , r ) = ( d m n + r , ( m - 1 ) n + k + r ) n , k ∈ ℕ . It is known that the ( m , r ) -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = ( d , h ) with h ( 0 ) = 0 and d ( 0 ) , h ' ( 0 ) ≠ 0 , we obtain the generating function of its ( m , r ) -central coefficients and give an explicit representation for the ( m , r ) -central Riordan array G ( m , r ) in terms of the Riordan array G . Meanwhile, the...

A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

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 p k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let p * k , m ( j , n ) 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 p * k , m ( j , n ) = p k , m ( j , n ) .

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