We consider $(k,j)$-colored partitions, partitions in which $k$ colors exist but at most $j$ colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where $k$ and $j$ are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the number of partitions in the $N \times M$ box with a given number of part sizes, and extend to multiple colors a conjecture of Dousse and Kim on unimodality in overpartitions.
We derive an explicit formula for a restricted partition function P_n^m(s) with constraints making use of known expression for a restricted partition function W_m(s) without constraints
In this note, we provide a simple derivation of expressions for the restricted partition function and its polynomial part. Our proof relies on elementary algebra on rational functions and a lemma that expresses the polynomial part as an average of the partition function.
We use the 0-1 tableaux as a tool for enumerating permutations and partitions with restricted minima. The method used is extended for permutation pairs and partition pairs generated by a bipartite 0-1 tableaux.
Let $\mathbf a=(a_1,\ldots,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.
In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with the restriction that the size of each block is contained in a given set. One of the main applications is in the study of lonesum matrices.
For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ and its polynomial part.
Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.
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.
In this paper, we combined two types of partitions and introduced 2-colored Rogers-Ramanujan partitions. By finding some functional equations and using a constructive method, some identities have been found. Some Overpartition identities coincide with our findings. A correspondence between colored partitions and overpartitions is provided.
We consider the following type of question: Given a partial proper $d$-edge coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors for the non-colored edges of $Q_d$,can we extend the partial coloring to a proper $d$-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.
Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.
In 2009, Blagojevic, Matschke & Ziegler established the first tight colored Tverberg theorem, but no lower bounds for the number of colored Tverberg partitions. We develop a colored version of our previous results (2008), and we extend our results from the uncolored version: Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. This follows from similar results on the number of colored Birch partitions.
Generalizing Reiner's notion of set partitions of type $B_n$, we define colored $B_n$-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored $B_n$-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored $B_n$-partition. We find an asymptotic expression of the total number of colored $B_n$-partitions up to an error of $O(n^{-1/2}\log^{7/2}{n})$, and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored $B_n$-pa…
Agarwal introduced $n$-color compositions in 2000 and most subsequent research has focused on restricting which parts are allowed. Here we focus instead on restricting allowed colors. After three general results, giving recurrence formulas for the cases of given allowed colors, given prohibited colors, and colors satisfying modular conditions, we consider several more specific conditions, establishing direct formulas and connections to other combinatorial objects. Proofs are combinatorial, mostly using the notion of spotted tilings introduced by the first named author in 2012.
We prove a conjecture due independently to Yan and Martinez--Savage that asserts inversion sequences with no weakly decreasing subsequence of length $3$ and enhanced $3$-noncrossing partitions have the same cardinality. Our approach applies both the generating tree technique and the so-called obstinate kernel method developed by Bousquet-Mélou. One application of this equinumerosity is a discovery of an intriguing identity involving numbers of classical and enhanced $3$-noncrossing partitions.
A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored $4$-cycles. Given a $(d,s)$-edge colorable graph $G$ and a list assigment $L$ of forbidden colors for the edges of $G$ satisfying certain sparsity conditions, we prove that there is a proper $d$-edge coloring of $G$ that avoids $L$, that is, a proper edge coloring $φ$ of $G$ such that $φ(e) \notin L(e)$ for every edge $e$ of $G$.
We generalize overpartitions to (k,j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. We find connections to divisor sums, the Han/Nekrasov-Okounkov hook length formula and a possible approach to a finitization, and other topics, suggesting that a rich mine of results is available.
We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a unification of k-colored partitions and overpartitions. We establish the observed relations at the constant and linear terms for all n, and for j=2 in their quadratic term, with the associated hook/frequency identities. Further results of this type seem likely.
Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities
College of Mathematics and Statistics & Center for Discrete Mathematics, Chongqing University（重庆大学数学与统计学院与离散数学中心）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
We derive an asymptotic formula for $A(n,j,r)$ the number of integer partitions of $n$ into at most $j$ parts each part $\le r$. We assume $j$ and $r$ are near their mean values. We also investigate the second largest part, the number of parts $\ge 2$, etc. We show that the fraction of the partitions of an even integer $n$ that are graphical, ie. whose parts form the degree sequence of a simple graph, is $O(\ln^{-1/2} n)$. Probabilistic results are used in our discussion of graphical partitions. The George Szekeres circle method is essential for our asymptotic results on partitions. We determi…
For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π^{\bullet}_{\vec{c}}$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition $\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^{B}$ where $B$ is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homol…
Enumeration of pattern-avoiding objects is an active area of study with connections to such disparate regions of mathematics as Schubert varieties and stack-sortable sequences. Recent research in this area has brought attention to colored permutations and colored set partitions. A colored partition of a set $S$ is a partition of $S$ with each element receiving a color from the set $[k]=\{1,2,...,k\}$. Let $Π_n\wr C_k$ be the set of partitions of $[n]$ with colors from $[k]$. In an earlier work, the authors study pattern avoidance in colored set partitions in the equality sense. Here we study p…
Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call \emph{$r$-colored set partitions}). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the …
We consider the graph coloring game, a game in which two players take turns properly coloring the vertices of a graph, with one player attempting to complete a proper coloring, and the other player attempting to prevent a proper coloring. We show that if a graph $G$ has a proper coloring in which the game coloring number of each bicolored subgraph is bounded, then the game chromatic number of $G$ is bounded. As a corollary to this result, we show that for two graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product $G_1 \square G_2$ has bounded game chromatic number, ans…
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 $n$?
The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$
P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can be separated along the colors. In this paper, we show a more colorful theorem.
A {\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever the edges of the graph $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $p$ vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an $r$-edge-colored complete graph.
Pattern avoidance in the symmetric group $S_n$ has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to $S_n\wr C_c$, the objects of which can be viewed as "colored permutations". Another body of research that has grown from the study of pattern avoidance in permutations is pattern avoidance in $Π_n$, the set of set partitions of $[n]$. Pattern avoidance in set partitions is a generalization of the well-studied notion of noncrossing partitions. Motivated by recent results in pattern a…
A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with $k$ colors. A parameter, namely $Φ_k (G)$, which is the number of non-equivalent distinguishing colorings of a graph $G$ with at most $k$ colors, is shown here to have an application in calculating the distinguishing number of the lexicographic product and the $X$-join o…
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an…
For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions.
We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding ce…
We show that for any fixed integer $m \geq 1$, a graph of maximum degree $Δ$ has a coloring with $O(Δ^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(Δ^{3/2})$ colors suffice, and acyclic colorings, for which $O(Δ^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph o…
Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for $p_{-k}(n)$ with $k=2, 6$, and $7$. For example, for all integers $n\geq0$ and $α\geq1$, we prove that \begin{align*} p_{-2}\left(5^{2α-1}n+\dfrac{7\times5^{2α-1}+1}{12}\right) &\equiv0\pmod{5^α} \end{align*} and \begin{align*} p_{-2}\left(5^{2α}n+\dfrac{11\times5^{2α}+1}{12}\right) &\equiv0\pmod{5^{α+1}}. \end{align*}
A generalized crank ($k$-crank) for $k$-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$-crank. Namely, we first obtain some inequalities between the $k$-crank counts $M_{k}(r,m,n)$ for $m=2,3$ and $4$, then we prove the positivity of symmetrized even $k$-crank moments weighted by the parity for $k=2$ and $3$. We conclude with several remarks on furthering the study initiated here.
Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy's identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.
Andrews, Brietzke, Rødseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$ and thus confirm the conjecture of Andrews, et al.
In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note, we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors.
A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of interest to this work. We first characterize the restricted growth functions of several avoidance classes based on partitions of size four, enumerate these avoidance classes, and consider the distribution of the Wachs and White statistics across these avoidance classes. We also investigate the equidistribution of statistics between avoidance classes based on mu…
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".
We discuss coloring and partitioning questions related to Sperner's Lemma, originally motivated by an application in hardness of approximation. Informally, we call a partitioning of the $(k-1)$-dimensional simplex into $k$ parts, or a labeling of a lattice inside the simplex by $k$ colors, "Sperner-admissible" if color $i$ avoids the face opposite to vertex $i$. The questions we study are of the following flavor: What is the Sperner-admissible labeling/partitioning that makes the total area of the boundary between different colors/parts as small as possible? First, for a natural arrangement of…
Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ with $k=1, 3$, and $9$. For example, for all integers $n\geq0$ and $α\geq1$, we prove that \begin{align*} p_{3,3}\left(3^αn+\dfrac{3^α+1}{2}\right) &\equiv0\pmod{3^{α+1}} \end{align*} and \begin{align*} p_{3,3}\left(3^{α+1}n+\dfrac{5\times3^α+1}{2}\right) &\equiv0\pmod{3^{α+4}}. \end{align*}
We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous results for $b$-colored partitions, where each part occurs in $b$ colors; for $b$-colored partitions with odd parts (or distinct parts); for partitions where the part $k$ comes in $k$ colors; and, overpartitions.
Inspired by Andrews' 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.
Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on partitions with restricted parts, which were confirmed analytically by Andrews and Chern and combinatorially by Yang. Analogous to Euler's partition theorem, it is known that the number of compositions of $n$ with odd parts equals the number of compositions of $n+1$ with parts greater than one, as both numbers equal the Fibonacci number $F_n$. Recently, Sills provid…
Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color. Builder's goal is to locate a particular monochromatic structure in Painter's coloring by revealing the color of as few edges as possible. The fewest number of turns required for Builder to win this game is known as the restricted online Ramsey number. In this paper, we consider the situation where this "particular monochromatic structure" is a large matching or …
