Relations binaires entre partitions
Remarks to a modification of Ramsey-type theorems
Representation functions with different weights
For any given positive integer k, and any set A of nonnegative integers, let denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both and hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.
Rogers-Ramanujan-Slater type identities.
RSK insertion for set partitions and diagram algebras.
Schur's determinants and partition theorems.
Self-conjugate vector partitions and the parity of the spt-function
Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. Recently, we found new combinatorial interpretations of congruences for the spt-function modulo 5 and 7. These interpretations were in terms of a restricted set of weighted vector partitions which we call S-partitions. We prove that the number of self-conjugate S-partitions, counted with a certain weight, is related to the coefficients of a certain mock theta function studied by the first author,...
Sets of parts such that the partition function is even
Shift-induced dynamical systems on partitions and compositions.
Simultaneous maj statistics.
Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals.
Some arithmetic properties of overpartition -tuples.
Some congruences for 3-component multipartitions
Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).
Some finite congruence lattices, I
Some finite generalizations of Euler's pentagonal number theorem
Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem.
Some formulas that enumerate certain partitions and graphs
Some general problems on the number of parts in partitions
Some new infinite families of congruences modulo 3 for overpartitions into odd parts
Let denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for modulo 3. For example, we prove that for n, α ≥ 0, .
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...
Some -analogues of determinant identities which arose in plane partition enumeration.