Related papers: Compositions, Partitions, and Fibonacci Numbers
The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$…
In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.
Concave compositions are ordered partitions whose parts are decreasing towards a central part. We study the distribution modulo $a$ of the number of concave compositions. Let $c(n)$ be the number of concave compositions of $n$ having even…
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 cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions…
In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function…
We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b. This resolves a recent conjecture of Stanley and Zanello.
Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim…
Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).
The number of partitions of $n$ wherein odd parts are distinct and even parts are unrestricted, often denoted by $pod(n)$. In this paper, we provide linear recurrence relations for $pod(n)$, and the connections of $pod(n)$ with other…
In this paper, we give a bijective proof of the reduced lecture hall partition theorem. It is possible to extend this bijection in lecture hall partition theorem. And refined versions of each theorems are also presented.
An integer composition of a nonnegative integer $n$ is a tuple $(\pi_1,\ldots,\pi_k)$ of nonnegative integers whose sum is $n$; the $\pi_i$'s are called the parts of the composition. For fixed number $k$ of parts, the number of $f$-weighted…
For a given integer $d\ge 1$, we consider $\binom{n+d-1}{d}$-color compositions of a positive integer $\nu$ for which each part of size $n$ admits $\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such…
In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…
We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d,…
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…
In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and…
It is proved that the number of partitions of n with odd mex and k parts that aren't ones equals the number of partitions of n with nonnegative crank and k parts that aren't ones..
We study formulas expressing Fibonacci numbers as sums over compositions using free submonoids of the free monoid of compositions with parts 1 and 2.
Recently, Andrews gave a detailed study of partitions with even parts below odd parts in which only the largest even part appears an odd number of times. In this paper, we provide a combinatorial proof of the generating function identity of…