Vol. 19, No. 2, 1966

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Restricted bipartite partitions

L. Carlitz and David Paul Roselle

Vol. 19 (1966), No. 2, 221–228
Abstract

Let πk(n,m) denote the number of partitions

n = n1 + n2 + + nk
m = m1 + m2 + + mk
subject to the conditions «««< Updated upstream
min(nj,mj) ≧ max (nj+1,mj+1 ) (j = 1,2,⋅⋅⋅ ,k− 1).

Put

           ∑∞
ξ(k)(x,y) =     πk(n,m )xnym
n,m=0

We show that

 (k)       ∏k ---------1−-x2j−-1y2j−1---------
ξ  (x,y) =   (1− xjyj)(1 − xjyj−1)(1− xj−1yj),
j=1

 ∑∞           n  m            ∑∞  k (k)
π(n,m; λ)x  y  = 1+ (1− λ)   λ ξ  (x,y),
n,m=0                         k=1

 ∞∑                ∑∞
ψ(n,m)xnym =     xnynξ(n)(x2,y2),
n,m=0              n=0

where π(n,m;λ) denotes the number of “weighted” partitions of (n,m) and ψ(n,m) is the number of partitions into odd parts (nj, mj all odd).

Mathematical Subject Classification
Primary: 10.48
Milestones
Received: 20 March 1965
Published: 1 November 1966
Authors
L. Carlitz
David Paul Roselle