Combinatorics of integer partitions in arithmetic progression.
Computation of -partial fractions.
Set partitions with successions and separations.
-complementing subsets of nonnegative integers.
Labeled factorization of integers.
Recurrence relations and two-dimensional set partitions.
On the partitions of a number into arithmetic progressions.
Enumeration of partitions by long rises, levels, and descents.
Parity-alternating permutations and successions
The study of parity-alternating permutations of {1, 2, … n} is extended to permutations containing a prescribed number of parity successions - adjacent pairs of elements of the same parity. Several enumeration formulae are computed for permutations containing a given number of parity successions, in conjunction with further parity and length restrictions. The objects are classified using direct construction and elementary combinatorial techniques. Analogous results are derived for circular permutations....
Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial
We note that every positive integer N has a representation as a sum of distinct members of the sequence , where d(m) is the number of divisors of m. When N is a member of a binary recurrence satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.
Some Parity Statistics in Integer Partitions
We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands may be enumerated according to descents while tracking the individual parities of and . There are two types of parity levels, E = E and O = O, and four types of parity-descents, E > E, E > O, O > E and O > O, where E and O represent arbitrary even and odd summands. We obtain functional equations and explicit...
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