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PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

  • BERNARD L. S. LIN (a1)

Abstract

For positive integers $t_{1},\ldots ,t_{k}$ , let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$ ) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$ . Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$ . We establish a $q$ -series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

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This work was supported by the National Natural Science Foundation of China (No. 11871246), the Natural Science Foundation of Fujian Province of China (No. 2019J01328) and the Program for New Century Excellent Talents in Fujian Province University (No. B17160).

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[1] Andrews, G. E., The Theory of Partitions (Addison-Wesley, New York, 1976).
[2] Andrews, G. E., Beck, M. and Robbins, N., ‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc. 143 (2015), 42834289.
[3] Breuer, F. and Kronholm, B., ‘A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins’, Res. Number Theory 2 (2016), Article ID 2, 15 pages.
[4] Chapman, R., ‘Partitions with bounded differences between largest and smallest parts’, Australas. J. Combin. 64 (2016), 376378.
[5] Chern, S., ‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math. 47 (2017), 2326.
[6] Chern, S., ‘An overpartition analogue of partitions with bounded differences between largest and smallest parts’, Discrete Math. 340 (2017), 28342839.
[7] Chern, S. and Yee, A. J., ‘Overpartitions with bounded part differences’, European J. Combin. 70 (2018), 317324.
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