Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the forbidden 0-1 matrix $P$ if $A$ avoids $P$ but changing any zero to a one in $A$ creates a copy of $P$. Define $sat(n, P)$ to be the minimum possible number of ones in an $n \times n$ 0-1 matrix that is saturating for $P$. Fulek and Keszegh proved that for every 0-1 matrix $P$, either $sat(n, P) = O(1)$ or $sat(n, P) = Θ(n)$. They found two 0-1 matrices $P$ for which $sat(n, P) = O(1)$, as well as infinite families of 0-1 matrices $P$ for which $sat(n, P) = Θ(n)$. Their results imply that $sat(n, P) = Θ(n)$ for almost all $k \times k$ 0-1 matrices $P$.
Fulek and Keszegh conjectured that there are many more 0-1 matrices $P$ such that $sat(n, P) = O(1)$ besides the ones they found, and they asked for a characterization of all permutation matrices $P$ such that $sat(n, P) = O(1)$. We affirm their conjecture by proving that almost all $k \times k$ permutation matrices $P$ have $sat(n, P) = O(1)$. We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions.
Gilbert Strang posited that a permutation matrix of bandwidth $w$ can be written as a product of $N < 2w$ permutation matrices of bandwidth 1. A proof employing a greedy ``parallel bubblesort'' algorithm on the rows of the permutation matrix is detailed and further points of interest are elaborated.
For a permutation $π$, and the corresponding permutation matrix, we introduce the notion of {\em discrete derivative}, obtained by taking differences of successive entries in $π$. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.
Recently, Jelínek conjectured that there exists a bijection between certain restricted permutations and Fishburn matrices such that the bijection verifies the equidistribution of several statistics. The main objective of this paper is to establish such a bijection.
A 0-1 matrix $M$ contains a 0-1 matrix pattern $P$ if we can obtain $P$ from $M$ by deleting rows and/or columns and turning arbitrary 1-entries into 0s. The saturation function $\mathrm{sat}(P,n)$ for a 0-1 matrix pattern $P$ indicates the minimum number of 1s in a $n \times n$ 0-1 matrix that does not contain $P$, but changing any 0-entry into a 1-entry creates an occurrence of $P$. Fulek and Keszegh recently showed that the saturation function is either bounded or in $Θ(n)$. Building on their results, we find a large class of patterns with bounded saturation function, including both infinit…
We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.
We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the inversion words of permutations and the "usual" representation of descending plane partitions as families of non--intersec\-ting lattice paths.
The concept of S-permutation matrix is considered. A general formula for counting all disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the positive integer $n$ is formulated and proven in this paper. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of $n^2 \times n^2$ S-permutation matrices, it is sufficient to obtain some numerical characteristics of all $n\times n$ bipartite graphs.
A generalized Catalan matrix $(a_{n,k})_{n,k\ge 0}$ is generated by two seed sequences $\mathbf{s}=(s_0,s_1,\ldots)$ and $\mathbf{t}=(t_1,t_2,\ldots)$ together with a recurrence relation. By taking $s_\ell=2\ell+1$ and $t_\ell=\ell^2$ we can interpret $a_{n,k}$ as the number of partial permutations, which are $n\times n$ $0,1$-matrices of $k$ zero rows with at most one $1$ in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on tw…
We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of permutations and the (well-known) representation of descending plane partitions as families of non--intersecting lattice paths. (Taking a short detour, we will also exhibit how the (well--known) enumeration of descending plane partitions follows easily from the evaluation of Andrew's determinant.)
In this paper we prove that two quantities relating to the length of permutations defined on trees are independent of the structures of trees. We also find that these results are closely related to the results obtained by Graham and Pollak (Bell System Tech. J. 50(1971) 2495--2519) and by Bapat, Kirkland, and Neumann (Linear Alg. Appl. 401(2005) 193--209).
The combined work of Bousquet-Mélou, Claesson, Dukes, Jelínek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent s…
Orthogonal matrices which are linear combinations of permutation matrices have attracted enormous attention in quantum information and computation. In this paper, we provide a complete parametric characterization of all complex, real and rational orthogonal permutative matrices of order $4.$ We show that any such matrix can always be expressed as a linear combination of up to four permutation matrices. Finally we determine several matrix spaces generated by linearly independent permutation matrices such that any orthogonal matrix in these spaces is always permutative or direct sum of orthogona…
We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results …
Given two $k\times n$ matrices $A$ and $B$, we describe a couple of methods to solve the matrix equation $XA=BY$, where $X$ is an invertible $k\times k$ matrix, and $Y$ is an $n\times n$ permutation matrix, both of which we want to determine. We are interested in pursuing those techniques that have algebraic geometric flavor. An application to solving such a matrix equation comes from the cryptanalysis of McEliece cryptosystem. By using codewords of minimum weight of a linear code, in concordance with these methods of solving $XA=BY$, we present an efficient way to determine the entire encrypt…
We define and investigate a family of permutations matrices, called shuffling matrices, acting on a set of $N=n_1\cdots n_m$ elements, where $m\geq 2$ and $n_i\geq 2$ for any $i=1,\ldots, m$. These elements are identified with the vertices of the $m$-th level of a rooted tree with branch indices $(n_1,\ldots, n_m)$. Each of such matrices is induced by a permutation of $Sym(m)$ and it turns out that, in the case in which one considers the cyclic permutation $(1\ \ldots\ m)$, the corresponding permutation is the classical perfect shuffle. We give a combinatorial interpretation of these permutati…
The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation matrices. More specifically, we count the number of permutations matrices which are equal to their partial transpose and the number of permutation matrices whose partial transpose is still a permutation. We solve these problems also when restricted to symmetric permutation matrices only.
We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form $P+T$, where $P$ is a permutation matrix and $T$ has four non-zero entries, forming a square with entries $1$ and $-1$ in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.
The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics.
We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. We show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs, results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders…
Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify a subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new par…
The concept of S-permutation matrix is considered in this paper. It defines when two binary matrices are disjoint. For an arbitrary $n^2 \times n^2$ S-permutation matrix, a lower band of the number of all disjoint with it S-permutation matrices is found. A formula for counting a lower band of the number of all disjoint pairs of $n^2 \times n^2$ S-permutation matrices is formulated and proven. As a consequence, a lower band of the probability of two randomly generated S-permutation matrices to be disjoint is found. In particular, a different proof of a known assertion is obtained in the work. T…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We illustrate this approach with applications to moments of orthogonal polynomials, permutations, signed permutations, and tableaux.
We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into irreducibles. The multiplicities involved have a nice combinatorial interpretation. We also generalize known results on asymptotic behavior of the conjugacy representation of $S_n$.
We present a proof of a conjecture about the relationship between Baxter permutations and pairs of alternating sign matrices that are produced from domino tilings of Aztec diamonds. It is shown that if and only if a tiling corresponds to a pair of ASMs that are both permutation matrices, the larger permutation matrix corresponds to a Baxter permutation. There has been a thriving literature on both pattern-avoiding permutations of various kinds and tilings of regions using dominos or rhombuses as tiles. However, there have not as of yet been many links between these two areas of enumerative com…
This work examines the concept of S-permutation matrices, namely $n^2 \times n^2$ permutation matrices containing a single 1 in each canonical $n \times n$ subsquare (block). The article suggests a formula for counting mutually disjoint pairs of $n^2 \times n^2$ S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of $n \times n$ binary matrices. The paper describe an algorithm that solves the main problem. To do that, every $n\times n$ binary matr…
A $d$-dimensional matrix is called \emph{$1$-polystochastic} if it is non-negative and the sum over each line equals~$1$. Such a matrix that has a single $1$ in each line and zeros elsewhere is called a \emph{$1$-permutation} matrix. A \emph{diagonal} of a $d$-dimensional matrix of order $n$ is a choice of $n$ elements, no two in the same hyperplane. The \emph{permanent} of a $d$-dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal. For a given order $n$ and dimension $d$, the set of $1$-polystochastic matrices forms a convex polytope that include…
Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general $n^2 \times n^2$ case are discussed in this paper. All bipartite graphs of the type $g=<R_g \cup C_g, E_g>$, where $|R_g |=|C_g |=2$ or $|R_g |=|C_g |=3$ are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the $4 \times 4$ and $9 \times 9$ cases are calculated.
We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet $\{0, 1, ..., k\}$, $k \geq 1$, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of $k$-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Ho…
Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $\log^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d}$ converges weakly to the circular law in probability as $n \to \infty$.
This paper introduces a polynomial blind algorithm that determines when two square matrices, $A$ and $B$, are permutation similar. The shifted and translated matrices $(A+βI+γJ)$ and $(B+βI+γJ)$ are used to color the vertices of two square, edge weighted, rook's graphs. Then the orbits are found by repeated symbolic squaring of the vertex colored and edge weighted adjacency matrices. Multisets of the diagonal symbols from non-permutation similar matrices are distinct within a few iterations, typically four or less.
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
Consider the regular representation of the sum over all permutations weighted by the sum of their descent, inversion, and fixed point multinomials. We compute the spectrum and the multiplicities of its elements of that matrix. Note that those multinomial statistics allow to apply the result on several permutation statistics like the number of fixed points, of descents, of inversions, and the major index at the same time.
A $d$-dimensional zero-one matrix $A$ avoids another $d$-dimensional zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeroes. Let $f(n,P,d)$ denote the maximum number of ones in a $d$-dimensional $n \times \cdots \times n$ zero-one matrix that avoids $P$. Fox proved for $n$ sufficiently large that $f(n, P, 2) = 2^{k^{Θ(1)}}n$ for almost all $k \times k$ permutation matrices $P$. We extend this result by proving for $d \geq 2$ and $n$ sufficiently large that $f(n, P, d) = 2^{k^{Θ(1)}}n^{d-1}$ for almost all $d$-dimensional permutation matrices $P$ of…
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are either 0 or 1, by analogy with permutation matrices. In this paper, we prove a simple characterization of permutation triangles, originally conjectured by Glick. We will also prove some properties of alternating sign triangles.
A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding problem for the proposed class of permutation codes as a linear programming (LP) problem. The main feature of this class of permutation codes, called LP decodable permutation codes, is this LP decodability. It is demonstrated that the LP decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the c…
In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the integer $n$ naturally arises. This is an interesting combinatorial problem that deserves its consideration. The pres…
We study the shape of the Young diagram λassociated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n\to\infty the rows of λbehave like the eigenvalues of a certain random matrix (traceless Gaussian Unitary Ensemble) with d rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asy…
A 0-1 matrix $M$ is saturating for a 0-1 matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by flipping any number of its $1$-entries to $0$-entries, and changing any $0$-entry to $1$-entry of $M$ introduces a copy of $P$. Matrix $M$ is semisaturating for $P$ if changing any $0$-entry to $1$-entry of $M$ introduces a new copy of $P$, regardless of whether $M$ originally contains $P$ or not. The functions $ex(n;P)$ and $sat(n;P)$ are the maximum and minimum possible number of $1$-entries a $n\times n$ 0-1 matrix saturating for $P$ can have, respectively. Function $ssat(n…
The saturation theorem of [Knutson-Tao '99] concerns the nonvanishing of Littlewood-Richardson coefficients. In combination with work of [Klyachko '98], it implies [Horn '62]'s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization [Friedland '00] to majorized sums of Hermitian matrices. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmanni…
In 2005 J.L. Waldspurger proved the following theorem: given a finite real reflection group $W$, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations $id-w$. Shortly thereafter E. Meinrencken extended the result to affine Weyl groups. P.V. Bibikov and V.S. Zhgoon then gave a uniform proof for a discrete reflection group acting on a simply-connected space of constant curvature. In this paper we show that the Waldspurger and Meinrenken theorems of type A give a new perspective on the combinatorics of the symmetric group. In p…
This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classi…
In the literatures, it is well-known that Fossorier code has the girth among LDPC codes. In this paper, we introduce a new class of low-density parity-check (LDPC) codes, with higher girth than other previous constructed codes. Especially we proposed a new method to construct LDPC codes using non fixed shift permutation matrices and full based matrices with higher girth than codes constructed by Fossorier.
A 0-1 matrix $M$ contains another 0-1 matrix $P$ if some submatrix of $M$ can be turned into $P$ by changing any number of $1$-entries to $0$-entries. $M$ is $\mathcal{P}$-saturated where $\mathcal{P}$ is a family of 0-1 matrices if $M$ avoids every element of $\mathcal{P}$ and changing any $0$-entry of $M$ to a $1$-entry introduces a copy of some element of $\mathcal{P}$. The extremal function $\operatorname{ex}(n,\mathcal{P})$ and saturation function $\operatorname{sat}(n,\mathcal{P})$ are the maximum and minimum possible weight of an $n\times n$ $\mathcal{P}$-saturated 0-1 matrix, respectiv…
Some randomized algorithms, used to obtain a random $n^2 \times n^2$ Sudoku matrix, where $n$ is a natural number, is reviewed in this study. Below is described the set $Π_n$ of all $(2n) \times n$ matrices, consisting of elements of the set $\mathbb{Z}_n =\{ 1,2,\ldots ,n\}$, such that every row is a permutation. It is proved that such matrices would be particularly useful in developing efficient algorithms in generating Sudoku matrices. An algorithm to obtain random $Π_n$ matrices is presented in this paper. The algorithms are evaluated according to two criteria - probability evaluation, and…
Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. We first establish this construction for random matrices. We define the traffic distribution of random matrices, which is richer than the *-distribution of free probability. The knowledge of the individual traffic distribut…
The X-ray of a permutation is defined as the sequence of antidiagonal sums in the associated permutation matrix. X-rays of permutation are interesting in the context of Discrete Tomography since many types of integral matrices can be written as linear combinations of permutation matrices. This paper is an invitation to the study of X-rays of permutations from a combinatorial point of view. We present connections between these objects and nondecreasing differences of permutations, zero-sum arrays, decomposable permutations, score sequences of tournaments, queens' problems and rooks' problems.
We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even pol…
We define and enumerate two new two-parameter permutation families, namely, placements of a maximum number of non-attacking rooks on $k$ chained-together $n\times n$ chessboards, in either a circular or linear configuration. The linear case with $k=1$ corresponds to standard permutations of $n$, and the circular case with $n=4$ and $k=6$ corresponds to a three-person chessboard. We give bijections of these rook placements to matrix form, one-line notation, and matchings on certain graphs. Finally, we define chained linear and circular alternating sign matrices, enumerate them for certain value…
