[seqfan] Re: Partitions and Dice

N. J. A. Sloane njas at research.att.com
Sat Feb 21 22:52:10 CET 2009

Frank wrote:

> This leads me to wonder: is t he number of partitions of n into k
> parts, each <= k, always equal to the number of partitions of n-1 into
> k-1 parts, each <= k+1?

Yes, this follows from Theorem 1.5 of Andrews, Thy of Partitions:

# ptns of a-c into exactly b-1 parts none exceeding c   =
# ptns of a-b into exactly c-1 parts none exceeding b

(take b = c+1)

The proof is by giving an explicit bijection


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