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Partitions with fixed differences between largest and smallest parts. (English) Zbl 1328.11106

The authors prove the following closed formula for the generating functions \[ \sum_{n=1}^{\infty}p(n,t)q^n=\frac{q^{t-1}(1-q)}{(1-q^t)(1-q^{t-1})}-\frac{q^{t-1}(1-q)}{(1-q^t)(1-q^{t-1})(q;q)_t}+\frac{q^t}{(1-q^{t-1})(q;q)_t}, \] where \(t>0\) and \(p(n,t)\) enumerates the number of ordinary partitions of \(n\) with difference \(t\) between the largest and the smallest parts. A generalization of this result is given for a class of partitions with an arbitrary number of specified distances.

MSC:

11P84 Partition identities; identities of Rogers-Ramanujan type
05A17 Combinatorial aspects of partitions of integers

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References:

[1] [sloaneonlineseq] The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org, 2014.
[2] Andrews, George E., The theory of partitions, Cambridge Mathematical Library, xvi+255 pp. (1998), Cambridge University Press, Cambridge · Zbl 0996.11002
[3] Flajolet, Philippe; Sedgewick, Robert, Analytic combinatorics, xiv+810 pp. (2009), Cambridge University Press, Cambridge · Zbl 1165.05001
[4] Stanley, Richard P., Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics 49, xiv+626 pp. (2012), Cambridge University Press, Cambridge · Zbl 1247.05003
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