In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modules for the type $A_{2n}^{(2)}, D_{n+1}^{(2)}$ and $B_n^{(1)}$, and also to make the connection between the result stated in paper one and the representation theory.
A graph has a perfect partition if all its perfect matchings can be partitioned so that each part is a 1-factorization of the graph. Let $L_{rm, r}=K_{rm,rm}-mK_{r,r}$. We first give a formula to count the number of perfect matchings of $L_{rm, r}$, then show that $L_{6,1}$ and $L_{8,2}$ have perfect partitions.
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…
We show that, up to multiplication by a factor $\frac{1}{(cq;q)_{\infty}}$, the weighted words version of Capparelli's identity is a particular case of the weighted words version of Primc's identity. We prove this first using recurrences, and then bijectively. We also give finite versions of both identities.
We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be obtained from it by repeated shortening. Further, we generalize the results to partitions: every partition of the Hamming space into $1$-error-correcting codes can be embedded in a partition of a space of some larger dimension into $1$-perfect codes. For the partitions, the embedding length is close to the theoretical bound for the general case and optimal f…
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of Kotzig's canonical partition to a decomposition of general graphs with perfect matchings. It contains a short proof for the theorem of the canonical partition. These results give decompositions which are canonical, that is, unique to given graphs. We also show that there are correlations between these two and that these can be computed in polynomial time.
Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods invo…
We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.
In the first paper of this series, we gave infinite families of coloured partition identities which generalise Primc's and Capparelli's classical identities. In this second paper, we study the representation theoretic consequences of our combinatorial results. First, we show that the difference conditions we defined in our $n^2$-coloured generalisation of Primc's identity, which have a very simple expression, are actually the energy function with values in $\{0,1,2\}$ for the perfect crystal of the tensor product of the vector representation and its dual in $A_{n-1}^{(1)}$. Then we introduce a…
This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen-Macaulayness of a poset. Applications to complexes of injective words are also included.
We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generating from k-tuples of words for any fixed natural number k) into a partition theorem about co…
Let $S=(s_n)_{n\geq 1}$ be a sequence with elements in a commutative monoid $(\mathcal{M},+,0)$. In this paper, we provide an explicit formula for $$\sum_{\la} C(\la) q^{\sum_{n\geq 1} \la_n\cdot s_n}$$ where $\la=(\la_1,\ldots)$ run through some subsets of over-partitions, and $C(\la)$ is a certain product of ``colors'' assigned to the parts of $\la$, and $q^s$ is a formal power of $q$ for $s\in M$. This formula allows us not only to retrieve several known Schmidt-type theorems but also to provide new Schmidt-type theorems for non-periodic sequences $S$. For example, when $(M,+,0)=(\mathbb{Z}…
We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let $\op_{n,k}(p)$ be the number of ordered set partitions of $\{1,2,\ldots,n\}$ into $k$ blocks that avoid a permutation pattern $p$. We establish an explicit identity between the number $\op_{n,k}(p)$ and the numbers of words avoiding the inverse of $p$. This identity allows us to easily translate results on pattern-avoiding words obtained in earlier works into equivalent results on pattern-avoiding ordered set partitions. In particular, \emph{(a)} we dete…
A complete partition theory is presented for omega-located words (and omega-words), namely for located words over an infinite alphabet dominated by a fixed increasing sequence. This theory strengthens in an essential way the classical Carlson, Furstenberg-Katznelson, and Bergelson-Blass-Hindman partition theory for words over a finite alphabet. Consequences of this theory are strong simultaneous extensions of the classical Hindman, Milliken-Taylor partition theorem, and of a van der Waerden theorem for general semigroups, extending results of Hindman-Strauss and Beiglbock.
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under rotations. For positive integers k and n, we call a necklace (k,n)-perfect if each word of length k occurs exactly n times at positions which are different modulo n for any convention on the starting point. We call a necklace perfect if it is (k,k)-perfect for some k. We prove that every arithmetic sequence with difference coprime with the alphabet size induc…
The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.
Given non-negative weights w_S on the k-subsets S of a km-element set V, we consider the sum of the products w_{S_1} ... w_{S_m} for all partitions V = S_1 cup ... cup S_m into pairwise disjoint k-subsets S_i. When the weights w_S are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman-Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect mat…
We obtain limit shape of perfect matchings on a large class of rail-yard graphs with right boundary condition given by the empty partition, and left boundary condition given by either by a staircase partition with constant density or a piecewise partition with densities either 1 or 0. We prove the parametric equations for the frozen boundary, and find conditions under which the frozen boundary is a cloud curve, or a union of disjoint cloud curves.
Alladi and Gordon introduced the method of weighted words in 1993 to prove a refinement and generalisation of Schur's partition identity. Together with Andrews, they later used it to refine Capparelli's and Göllnitz' identities too. In this paper, we present a new variant of this method, which can be used to study more complicated partition identities, and apply it to prove refinements and generalisations of three partition identities. The first one, Siladić's theorem (2002), comes from vertex operator algebras. The second one, a conjectural identity of Primc (1999), comes from crystal base th…
An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(σ)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $σ$ and obtained the formula for $|\mathcal{OP}_{n,k}(σ)|$ when the pattern $σ$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|\mathcal{OP}_{n,k}(σ)|$ when the pattern $σ$ is of length $3$. In this paper, we define a new patt…
We study packing densities for set partitions, which is a generalization of packing words. We use results from the literature about packing densities for permutations and words to provide packing densities for set partitions. These results give us most of the packing densities for partitions of the set $\{1,2,3\}$. In the final section we determine the packing density of the set partition $\{\{1,3\},\{2\}\}$.
We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where $t_{G}\Spec(G) \subseteq \ZZπ$, and $H$ is a circulant with odd eigenvalues, their weak product $G \times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a g…
We revisit the following problem (along with its higher dimensional variant): Given a set $S$ of $n$ points inside an axis-parallel rectangle $U$ in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in $U$ but contains no points of $S$. (I) We present an algorithm that finds a large empty box amidst $n$ points in $[0,1]^d$: a box whose volume is at least $\frac{\log{d}}{4(n + \log{d})}$ can be computed in $O(n+d \log{d})$ time. (II) To better analyze the above approach, we introduce the concepts of perfect vector sets and properly overlapping partitions, in connectio…
Let $H$ be a $3$-partite $3$-uniform hypergraph, i.e. a $3$-uniform hypergraph such that every edge intersects every partition class in exactly one vertex, with each partition class of size $n$. We determine a Dirac-type vertex degree threshold for perfect matchings in $3$-partite $3$-uniform hypergraphs.
Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if $K\subseteq B$ and $S\subseteq W$. A Clique-Stable Set Separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique-Stable Set Separator cont…
Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $t\geq 1$, denote by $\mathcal{MG}_t$ the class of graphs whose maximum multiplicity is at most $t$. A graph $G$ is called strictly $t$-degenerate if every non-empty subgraph $H$ of $G$ contains a vertex $v$ whose degree in $H$ is at most $t-1$. The point partition number $χ_t(G)$ of $G$ is smallest number of colors needed to color the vertices of $G$ so that each vertex receives a color and vertices with the same color induce a strictly $t$-degenerate subgraph of $G$. So $χ_1$ is …
The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition intersection graph to characterize when a character set has a unique perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition intersection graph to find a maximum compatible set of …
A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the wei…
A graph $G$ is said to be $2$-divisible if for all (nonempty) induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $ω(A) < ω(H)$ and $ω(B) < ω(H)$. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $H[A]$ is perfect and $ω(B) < ω(H)$. We prove that if a graph is $(P_5,C_5)$-free, then it is $2$-divisible. We also prove that if a graph is bull-free and either odd-hole-free or $P_5$-free, then it is perfectly divisible.
We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depen…
An extended $1$-perfect trade is a pair $(T_0,T_1)$ of two disjoint binary distance-$4$ even-weight codes such that the set of words at distance $1$ from $T_0$ coincides with the set of words at distance $1$ from $T_1$. Such trade is called primary if any pair of proper subsets of $T_0$ and $T_1$ is not a trade. Using a computer-aided approach, we classify nonequivalent primary extended $1$-perfect trades of length $10$, constant-weight extended $1$-perfect trades of length $12$, and Steiner trades derived from them. In particular, all Steiner trades with parameters $(5,6,12)$ are classified.
Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee co…
A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a "combinatorial" polynomial-time algorithm that yields an optimal colouring of a perfect graph. A skew partition in $G$ is a partition $(A,B)$ of $V(G)$ such that $G[A]$ is not connected and $\bar{G}[B]$ is not connected, where $\bar{G}$ denotes the complement graph…
In a recent paper, we generalized a partition identity stated by Siladić in his study of the level one standard module of type $A_2^{(2)}$. The proof used weighted words with an arbitrary number of primary colors and all the secondary colors obtained from these primary colors, and a brand new variant of the bijection of Bressoud for Schur's partition identity. In this paper, the first of two, we analyze this variant of Bressoud's algorithm in the framework of statistical mechanics, where an integer partition is viewed as an amount of energy shared, according to certain properties, between seve…
We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\R^2$. In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of $Λ$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^…
In this paper, we deal with a simple geometric problem: Is it possible to partition a rectangle into $k$ non-congruent rectangles of equal area? This problem is motivated by the so-called `Mondrian art problem' that asks a similar question for dissections with rectangles of integer sides. Here, we generalize the Mondrian problem by allowing rectangles of real sides. In this case, we show that the minimum value of $k$ for a rectangle to have a `perfect Mondrian partition' (that is, with non-congruent equal-area rectangles) is seven. Moreover, we prove that such a partition is unique (up to symm…
