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On self-conjugate split \(n\)-color partitions. (English) Zbl 1495.05015

Summary: Analogous to the definition of self-conjugate \(n\)-color partitions, we introduce here self-conjugate split \(n\)-color partitions. The self-conjugate split \(n\)-color partitions arise as a modification of another, existing class of partitions. In addition to the generating function and recurrence relation of self-conjugate split \(n\)-color partitions, we find several combinatorial identities which associate these partitions with other combinatorial structures. We give a bijection from the set of split \(n\)-color partitions of a positive integer \(\nu\) onto that of partitions of \(\nu\) with “\(\begin{pmatrix} n+1 \\ 2 \end{pmatrix}\) copies of \(n\)”. Moreover, an explicit bijection between the set of restricted self-conjugate split \(n\)-color partitions of \(\nu\) and the set of restricted \(n\)-color partitions of \(\nu\) has been constructed. Some results involving new restricted split \(n\)-color partition functions are also obtained.

MSC:

05A15 Exact enumeration problems, generating functions
05A17 Combinatorial aspects of partitions of integers
05A19 Combinatorial identities, bijective combinatorics
11P81 Elementary theory of partitions
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