Some classes of numbers and derivatives.
Some combinatorial identities.
Some combinatorial problems, connected with product-isomorphisms of binary relations
Some combinatorial results about the operators with jumping nonlinearities
Some combinatorics involving ξ-large sets
We prove a version of the Ramsey theorem for partitions of (increasing) n-tuples. We derive this result from a version of König's infinity lemma for ξ-large trees. Here ξ < ε₀ and the notion of largeness is in the sense of Hardy hierarchy.
Some computational formulas for -Nörlund numbers.
Some congruence properties of binomial coefficients and linear second order recurrences.
Some congruences concerning the Bell numbers.
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 congruences involving binomial coefficients
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that , where the central trinomial coefficient Tₙ is the constant term in the expansion of . We also prove three congruences modulo p³ conjectured by Sun, one of which is . In addition, we get some new combinatorial identities.
Some Considerations of Fibonacci Permutations
Some debts I owe.
Some easily derivable integer sequences.
Some equidistributed statistics on Genocchi permutations.
Some equinumerous pattern-avoiding classes of permutations.
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 for the central trinomial and Motzkin numbers.
Some formulas that enumerate certain partitions and graphs
Some further results on Legendre numbers.