Andrews, George E.; Beck, Matthias; Robbins, Neville Partitions with fixed differences between largest and smallest parts. (English) Zbl 1328.11106 Proc. Am. Math. Soc. 143, No. 10, 4283-4289 (2015). 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. Reviewer: Mircea Merca (Cornu) Cited in 3 ReviewsCited in 11 Documents MSC: 11P84 Partition identities; identities of Rogers-Ramanujan type 05A17 Combinatorial aspects of partitions of integers Keywords:integer partition; fixed difference between largest and smallest parts; rational generating function; quasipolynomial Software:OEIS PDFBibTeX XMLCite \textit{G. E. Andrews} et al., Proc. Am. Math. Soc. 143, No. 10, 4283--4289 (2015; Zbl 1328.11106) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Triangular numbers repeated. a(n) = n - d(n), where d(n) is the number of divisors of n (A000005). Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k. Triangle read by rows: T(n,k) is number of partitions of n that have k parts smaller than the largest part (n>=1, k>=0). Number of partitions p of n such that max(p)-min(p)=3. Number of partitions p of n such that max(p)-min(p) = 4. Number of partitions p of n such that max(p)-min(p) = 5. Number of partitions p of n such that max(p)-min(p) = 6. Number of partitions p of n such that max(p)-min(p) = 7. Number of partitions p of n such that max(p)-min(p) = 8. Number of partitions p of n such that max(p)-min(p) = 9. Number of partitions p of n such that max(p)-min(p) = 10. Triangle read by rows: T(n,k) is the difference between the largest and the smallest part of the k-th partition in the list of colexicographically ordered partitions of n, with n>=1 and 1<=k<=p(n), where p(n) is the number of partitions of n. 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.