TY - JOUR

T1 - Compositions with restricted parts

AU - Huang, Jia

N1 - Funding Information:
The author uses SageMath to discover and verify the main results in this paper. He is grateful to Goerge Beck, Toufik Mansour, and Augustine Munagi for pointing out some typos and useful references on compositions with restricted parts. He also thanks the anonymous referees for helpful comments and suggestions.
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/7

Y1 - 2020/7

N2 - Euler showed that the number of partitions of n into distinct parts equals the number of partitions of n into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of n with odd parts equals the number of compositions of n+1 with parts greater than one, as both numbers equal the Fibonacci number Fn. Recently, Sills provided a bijective proof for this result using binary sequences, and Munagi proved a generalization similar to Glaisher's result using the zigzag graphs of compositions. Extending Sills’ bijection, we obtain a further generalization which is analogous to Franklin's result. We establish, both analytically and combinatorially, two closed formulae for the number of compositions with restricted parts appearing in our generalization. We also prove some composition analogues for the conjectures of Beck.

AB - Euler showed that the number of partitions of n into distinct parts equals the number of partitions of n into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of n with odd parts equals the number of compositions of n+1 with parts greater than one, as both numbers equal the Fibonacci number Fn. Recently, Sills provided a bijective proof for this result using binary sequences, and Munagi proved a generalization similar to Glaisher's result using the zigzag graphs of compositions. Extending Sills’ bijection, we obtain a further generalization which is analogous to Franklin's result. We establish, both analytically and combinatorially, two closed formulae for the number of compositions with restricted parts appearing in our generalization. We also prove some composition analogues for the conjectures of Beck.

KW - Composition

KW - Euler's partition theorem

KW - Restricted parts

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U2 - 10.1016/j.disc.2020.111875

DO - 10.1016/j.disc.2020.111875

M3 - Article

AN - SCOPUS:85080037338

VL - 343

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 7

M1 - 111875

ER -