[seqfan] Re: Partitions and Dice

[N. J. A. Sloane]

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

Neil