This note reports on the number of s-partitions of a natural number n. In an s-partition each cell has the form $2^k-1$ for some integer k. Such partitions have potential applications in cryptography, specifically in distributed computations of the form $a^n$ mod m. The main contribution of this paper is a correction to the upper bound on the number of s-partitions presented by Bhatt. We will give a precise asymptotics for the number of such partitions for a given integer n.
Using the index theory of seaweed algebras, we explore various new integer partition statistics. We find relations to some well-known varieties of integer partitions as well as a surprising periodicity result.
I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on partitions that is associative with a natural grading thus defining a free associative algebra on the set of integer partitions. As an outcome of the computations, certain sets of integers appear that I call difference sets and the product of the integers in a difference set is an invariant for a family of sets of partitions. The main combinatorial objects u…
There is a familiar conjugate for integer partitions: transpose the Ferrers diagram, and a conjugate for integer compositions: transpose a Ferrers-like diagram. Here we propose a conjugate for set partitions and show that it interchanges # singletons and # adjacencies. Its restriction to noncrossing partitions cropped up in a 1972 paper of Kreweras. We also exhibit an analogous pairs of statistics interchanged by the composition conjugate.
We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results about core partitions. For small values of a, we give formulas for the number of (a,b)-core partitions corresponding to numerical semigroups. We also study the number of partitions with a given hook set.
We prove a general lemma about partitioning the vertex set of a graph into subgraphs of bounded degree. This lemma extends a sequence of results of Lovász, Catlin, Kostochka and Rabern.
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.
We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension.
To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add up to the same sum.
We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions.
An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the p…
Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions equivalent to the classical Foulkes conjecture. We establish some of the properties of this relation via the construction of explicit inclusion of modules whose characters correspond to the plethysms considered. We also prove some stability properties for the number of irreducible occurring in these modules as $n$ grows.
We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.
In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an interesting statement about triangular numbers, those positive integers which can be partitioned into consecutive numbers beginning at 1. For every partition of a triangular number n into consecutive numbers we can partition the sequence of numbers beginning at 1, adding up to n again, such that every part of this partition adds up to exactly one number of the chose…
An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the $5$-regular case and show that among the subgraphs of minimum degree at least $3$, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all $5$-regular Abe…
Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in \mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset $F$ of $S$ such that $S=F ^{-1} \{x \in S: x A \bigcap A\neq\emptyset\}$ and $|F |\leq n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.
The polytope of integer partitions of $n$ is the convex hull of the corresponding $n$-dimensional integer points. Its vertices are of importance because every partition is their convex combination. Computation shows intriguing features of $v(n),$ the number of the polytope vertices: its graph has a saw-toothed shape with the highest peaks at prime $n$'s. We explain the shape of $v(n)$ by the large number of partitions of even $n$'s that were counted by N. Metropolis and P. R. Stein. These partitions are convex combinations of two others. We reveal that divisibility of $n$ by 3 also reduces the…
The subjects of rook equivalence and Wilf equivalence have both attracted considerable attention over the last half-century. In this paper we introduce a new notion of Wilf equivalence for integer partitions, and, using this notion, we prove that rook equivalence implies Wilf equivalence. We also prove that if we refine the notions of rook and Wilf equivalence in a natural way, then these two notions coincide. In [6] we prove that Wilf equivalence implies rook equivalence.
We study the rate of growth of $p(n,S,M)$, the number of partitions of $n$ whose parts all belong to $S$ and whose multiplicities all belong to $M$, where $S$ (resp. $M$) are given infinite sets of positive (resp. nonnegative) integers. We show that if $M$ is all nonnegative integers then $p(n,S,M)$ cannot be of only polynomial growth, and that no sharper statement can be made. We ask: if $p(n,S,M)>0$ for all large enough $n$, can $p(n,S,M)$ be of polynomial growth in $n$?
We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of partitions of $n$ is equal to the number of partitions of $2n+d{n \choose 2}$ with $n$ $d$-distant parts. We also provide a direct proof for this identity. This work is the result of our aim at finding a bijective proof for Rogers-Ramanujan identities.
Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give results on integer partitions with parts from numbers in modular arithmetic progression. This includes recursive formulas for the number of partitions using these modular parts. The triple product identity can derive further recursive formulas. Additionally, there is a recursive formula for the related sum of divisors function. The specific triple product iden…
It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain partitioning process that gives rise to a sequence of sets of "nested" partitions. We prove that at stage m, the cardinality of the resulting set will equal the m-th order Bell number. This set-theoretic interpretation enables us to make a natural definition of higher order Stirling numbers and to study the combinatorics of these entities. The cardinality of the ele…
The index of a Lie algebra is an important algebraic invariant. In 2000, Vladimir Dergachev and Alexandre Kirillov defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed algebra using a certain graph, so-called a meander. In a recent paper, Vincent Coll, Andrew Mayers, and Nick Mayers defined a new statistic for partitions, namely the index of a partition, which arises from seaweed Lie algebras of type A. At the end of their paper, they presented an interesting conjecture, which involves integer partitions into odd parts. Motiv…
We study filters in the partition lattice formed by restricting to partitions by type. The Möbius function is determined in terms of the easier-to-compute descent set statistics on permutations and the Möbius function of filters in the lattice of integer compositions. When the underlying integer partition is a knapsack partition, the Möbius function on integer compositions is determined by a topological argument. In this proof the permutahedron makes a cameo appearance.
We prove a conjecture of Drake and Kim: the number of $2$-distant noncrossing partitions of $\{1,2,...,n\}$ is equal to the sum of weights of Motzkin paths of length $n$, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.
A graph composition is a partition of the vertex set such that each member of the partition induces a connected sub- graph, and the composition number of a graph is the number of possible graph compositions. A partition of a set S of consecutive labelled vertices is said to have a minimax vertex v in S if the label of v is the smallest label in the set of all maximum labels over all members of the partition. This paper exhibits a recursive formula for the composition number of a certain class of graphs and estab- lishes a connection between the composition numbers of this class of graphs and t…
We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new `direct' proof of Rado's result.
Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting behavior of their zeros as sets and densities.
Let $X_1,\dots, X_n$ be independent integers distributed uniformly on $\{1,\dots, M\}$, $M=M(n)\to\infty$ however slow. A partition $S$ of $[n]$ into $ν$ non-empty subsets $S_1,\dots, S_ν$ is called perfect, if all $ν$ values $\sum_{j\in S_{\a}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $ν$. For $ν=2$, Borgs et al. proved, among other results, that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $κ:=\lim \tfrac{n}{\log M}>\tfrac{1}{\log 2}$, and that w.h.p. no perfect partition exists if $κ<\tfrac{1}{\log 2…
In a recent paper, Grau et al. (2017) studied the zero divisor graphs of the ring of Lipschitz integers modulo $n$, and computed the domination number of the undirected zero divisor graph of the ring of Lipschitz integers modulo $n$. But the case $n$ is a power of prime numbers remained open. In this note, this problem is solved. We also show the automorphism group of the zero divisor graph of the ring of Lipschitz integers modulo $2^s$.
We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to $1$. Amongst other things, they relate the number of $-1$s in the alternating sign trapezoids to certain elements in the column strict shifted plane partitions that generalise the notion of special parts in descending plane partitions. The advantage of these bijections is that they include configurations with $-1$s, which is a feature that many of the bijections in the realm of alternating sign arrays lack.
If $a_1, a_2, ..., a_k$ and $n$ are positive integers such that $n = a_1 + a_2 + ... + a_k$, then the sum $a_1 + a_2 + ... + a_k$ is said to be a \emph{partition of $n$} of \emph{length $k$}, and $a_1, a_2, ..., a_k$ are said to be the \emph{parts} of the partition. Two partitions that differ only in the order of their parts are considered to be the same. We say that two partitions \emph{intersect} if they have at least one common part. We call a set $A$ of partitions \emph{intersecting} if any two partitions in $A$ intersect. Let $P_{n,k}$ be the set of all partitions of $n$ of length $k$. We…
For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form $n=\sum_{a \in A} m_a a$ where $m_a \in M \cup {0}$ for all $a$, and all numbers $m_a$ but finitely many are 0. It is shown that there are infinite sets $A$ and $M$ so that $p(n,A,M)=1$ for every positive integer $n$. This settles (in a strong form) a problem of Canfield and Wilf. It is also shown that there is an infinite set $M$ and constants $c$ and $n_0$ so t…
In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.
The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove identities for counts of partitions with large rank in fixed residue classes.
The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.
Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial $f_λ(x)$ that we call the partition polynomial for the partition $λ$, with the aim to learn new properties of partitions. We prove a recursive formula for the derivatives of $f_λ(x)$ involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of $f_λ(x)$ is dense in $[0,1/2]$, pose a few open questions, and formulate a conjecture relating the integral to the length of…
A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of consecutive integers, or, more generally, whose parts form a finite arithmetic progression. This paper reviews the relation between trapezoidal numbers, partitions, and the set of divisors of a positive integer. There is also a complete proof of a theorem of Sylvester that produces a stratification of the partitions of an integer into odd parts and partitions int…
