Related papers: Even and odd compositions with restricted parts
It is well known that the number of partitions into distinct even parts equals the number of $4$-regular partitions. In this paper we prove identities relating certain restricted partitions into distinct even parts with restricted…
While the general form of even perfect numbers is well-known, the existence or non-existence of odd perfect numbers is still an open problem. We address this problem and prove that if a natural number is odd, then it's not perfect.
In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the…
Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called…
Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$ assert that $$ p(\ell n+\delta_{\ell})\equiv 0\pmod{\ell}, $$ where $0<\delta_{\ell}<\ell$ satisfies $24\delta_{\ell}\equiv 1\pmod{\ell}.$ By proving Subbarao's…
Inspired by OEIS sequence A377912, which consists of the nonnegative integers in which every even digit (except possibly the last) is immediately followed by a strictly larger digit, we define even-up and odd-up words over an alphabet of…
In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare…
A special case of an elegant result due to Anderson proves that the number of $(s,s+1)$-core partitions is finite and is given by the Catalan number $C_s$. Amdeberhan recently conjectured that the number of $(s,s+1)$-core partitions into…
We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m.…
The theory of Frobenius groups with Frobenius complements of even order largely reduces to tractable algebraic number theory. If we consider only Frobenius complements with an upper bound $s$ on the number of distinct primes dividing the…
We generalise Euler's partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan- Andrews-Gordon identities related to this theorem.
We prove combinatorially some identities related to Euler's partition identity (the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts). They were conjectured by Beck and proved by Andrews…
In this paper, the compositional inverses of a class of linearized permutation polynomials of the form $P(x)=x+x^2+\tr(\frac{x}{a})$ over the finite field $\mathbb{F}_{2^n}$ for an odd positive integer $n$ are explicitly determined.
Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…
In the first part we associate a periodic sequence to a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved
Carlitz considered integer compositions in which adjacent parts must be unequal. Arndt recently initiated the study of restricted compositions based on conditions applied to certain pairs of parts rather than to individual parts. Here, we…