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Andrews [‘Binary and semi-Fibonacci partitions’, J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$. We also give a new characterisation of semi-$m$-Fibonacci partitions and some congruences satisfied by the associated number sequence.
Toufik Mansour, Department of Mathematics Haifa University 31905 Haifa, Israel,
Augustine O. Munagi, The John Knopfmacher Centre for Applicable Analysis and Number Theory School of Mathematics University of the Witwatersrand Johannesburg 2050, South Africa
A descent in a permutation α1α2 · αn is an index i for which αi > αi+1. The number of descents in a permutation is a classical permutation statistic which was first studied by P. A. MacMahon almost a hundred years ago, and it still plays an important role in the study of permutations. Representing set partitions by equivalent canonical sequences of integers, we study this statistic among the set partitions, as well as the numbers of rises and levels. We enumerate set partitions with respect to these statistics by means of generating functions, and present some combinatorial proofs. Applications are obtained to new combinatorial results and previously-known ones.
A descent in a permutation α = α1α2 ··· αn is an index i for which αi > αi+1. The number of descents in a permutation is a classical permutation statistic. This statistic was first studied by MacMahon, and it still plays an important role in the study of permutation statistics. In this paper we study the statistics of numbers of rises, levels and descents among set partitions expressed as canonical sequences, defined below.
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