Enumerative combinatorics

We explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometric approach to answer it.  Our main object of study is the Erd\H{o}s-Szekeres tableau, or EST, of a number sequence.  An EST is the sequence of integral points whose coordinates record the length of the longest increasing and longest decreasing subsequence ending at each element of the sequence.  We define the  Order Poset of an EST in order to answer the question: What information about the sequence can be recovered by its EST?

We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct the corresponding bijections.

Diagonalizing a matrix A, that is finding two matrices P and D such that A = P DP −1 with D being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not need the second step when diagonalizing 2 × 2 matrices since they already appear as nonzero columns in A − λ i I, where λ i is the complementary eigenvalue. We also provide a shortcut method to find eigenvectors.

In previous publications ( J. Geom.Phys.38 (2001) 81-139 and references therein ) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23 ) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.

Let X be a random variable having a Poisson distribution
and mean . Using the unified generalizations of Stirling
numbers, a pair of generalizations for the nth factorial moment
E[(X)n] of X is defined. Through this, some generalizations for
the nth moment E[Xn] of X are established in terms of other
generalizations of the Stirling numbers of the second kind. Also,
relationships between these new generalized moments and some
well-known identities are mentioned.

(This seems to be the first few paragraphs all mashed together.
I hope the paper makes it here okay.)


Let Z denote the set of integers {.. . , −2, −1, 0, 1, 2,. . .} and let Z + denote the set of positive integers {1, 2,. . .}. Also let R denote the set of real numbers (also specified as all the numbers in the open interval (−∞, ∞)). For a ∈ Z + ∪ {0}, let N a = {1, 2,. .. , a}; when a = 0, N a is empty, i.e., N a = ∅ where ∅ denotes the empty set. Let S a be the set of all length-a sequences that are composed of distinct elements of N a ; these sequences are permutations of the elements of N a , thus S a is the set of (length-a) permutations of 1, 2,. .. , a. For example, S 3 = {1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2, 3 2 1}. Note we sometimes, as in this example, write sequences without commas separating the sequence elements. We have #S a = a! where a! = a(a − 1)(a − 2) · · · 2 · 1. (In general, we write #S to denote the number of elements in the set S, thus for example, #N a = a.) The symbol ! is called the factorial operator. We define 0! = 1. To construct an element of S a , we may choose the first element in a ways, and then the remaining sequence of a − 1 elements may be chosen in (a − 1)! ways, yielding a! ways overall. This is essentially a proof by induction, where the empty sequence comprising the set S 0 can be chosen in just one way. Note there is just 0! = 1 permutation of the empty set.

Egge, in a talk at the AMS Fall Eastern Meeting, 2012, conjectured that permutations avoiding the set of patterns {2143, 3142, τ }, where τ ∈ {246135, 254613, 524361, 546132, 263514}, are enumerated by the large Schröder numbers (and thus {2143, 3142, τ } with τ as above is Wilfequivalent to the set of patterns {2413, 3142}). Burstein and Pantone [J. Combin. 6 (2015) no. 1-2, 55-67] proved the case of τ = 246135. We prove the remaining four cases. As a byproduct of our proof, we also enumerate the case τ = 4132.

Two pairs of generalized q-factorial moments involving the Heine and the Euler distributions, respectively, are established. Moreover, these pairs of q-factorial moments are shown to be proper q-analogues of the generalized factorial moments.

Diagonalizing a matrix A, that is finding two matrices P and D such that A = PDP^-1 with D being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not need the second step when diagonalizing matrices with a spectrum, |σ(A)|≤ 2 since those vectors already appear as nonzero columns of the eigenmatrices, a term defined in this work. We further generalize this for matrices with |σ(A)|> 2 and show that eigenvectors lie in the column spaces of eigenmatrices of the complementary eigenvalues, an approach without using the classical Gauss-Jordan elimination of rows of a matrix. We introduce two major results, namely, the 2-Spectrum Lemma and the Eigenmatrix Theorem. As a conjecture, we further generalize the Jordan canonical forms for a new class of generalized eigenvectors that are produced by repeated multiples of certain eigenmatrices. We also provide several shortcut formulas to find eigenvectors that does no...

A rigorous presentation of the mathematical theory of enumeration methods and combinatoric colouring based on a group theoretical approach, ending with the celebrated theorem of Redfield-Polya. Designed for those who seek for a quick and rigorous introduction to some aspects of advanced combinatorics.

Diagonalizing a matrix $A$, that is finding two matrices $P$ and $D$ such that $A = PDP^{-1}$ with $D$ being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not need the second step when diagonalizing matrices with a spectrum, $\left|\sigma(A)\right|\leq 2$ since those vectors already appear as nonzero columns of the eigenmatrices, a term defined in this work. We further generalize this for matrices with $\left|\sigma(A)\right|> 2$ and show that eigenvectors lie in the column spaces of eigenmatrices, an approach without using the Gaussian elimination of rows of a matrix. We also provide several shortcut formulas to find eigenvectors that does not use echelon forms. The method can be summarized as ``Finding your puppy at your neighbors !"

We enumerate the independent sets of several classes of regular and al- most regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct the corresponding bijections.

This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference.

Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.

In this paper, we further develop the study of the translated Whitney numbers by deriving more combinatorial properties such as more recurrence relations, exponential and rational generating functions and the orthogonality and inverse relations. To achieve this goal, we introduce the ``signed'' translated Whitney numbers of the first kind. A relationship between these numbers and the Bernoulli polynomials is also briefly discussed.

Despite having been introduced in 1962 by C.L. Mallows, the combinatorial algorithm Patience Sorting is only now beginning to receive significant attention due to such recent deep results as the Baik-Deift-Johansson Theorem that connect it to fields including Probabilistic Combinatorics and Random Matrix Theory.

"We study generating functions for the number of permutations in $S_n$ subject to two restrictions. One of the restrictions belongs to $S_3$, while the other belongs to $S_k$. It turns out that in a large
variety of cases the answer can be expressed via Chebyshev polynomials of the second k"

We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between 1-hinge and 0-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that are known to be equidistributed via a complicated map to permutations that translates those to occurrences of certain patterns. We give two direct maps on tableaux that proves the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux.

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 certain pairs of 4-letter patterns and another pattern of arbitrary length.

In this paper, a q-analogue of the noncentral Whitney numbers of both kinds are define in terms of horizontal generating functions. Some properties such as recurrence relations, explicit formula, generating functions, orthogonality and inverse relations are established. Matrix
decomposition of these q-analogues is presented in an explicit and non-recursive form. Moreover, a q-analogue of the noncentral Dowling numbers and polynomials are defining and establish some of their properties.

We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between 1-hinge and 0-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that are known to be equidistributed via a complicated map to permutations that translates those to occurrences of certain patterns. We give two direct maps on tableaux that proves the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux.

The theory of {\Gamma}-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees.

We consider the set of permutations all of whose descents are from an even value to an even value. Proving a conjecture of Kitaev and Remmel, we show that these permutations are enumerated by Genocchi numbers, hence equinumerous to Dumont permutations of the rst and second kind, and thus may be called Dumont permutations of the third kind. We also dene the related Dumont permutations of the fourth kind.

In this paper, we obtain closed formulas for the number of reachable vertices in labelled plane trees by paths lengths, sinks, leaf sinks, first children, left most path, non-first children, and non-leaves. Our counting objects are plane trees having their edges oriented from a vertex of lower label towards a vertex of higher label. For each statistic, we obtain the average number of reachable vertices. Moreover, we obtain a counting formula for the number of plane trees on n vertices such that exactly k ≤ n are reachable from the root.

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize the RNA matrix but are of independent interest.

The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements–all in analogy with formulas for finite sets (which are the special case of q = 1). A direct-sum decomposition of a finite vector space is the vector space analogue of a set partition. This paper uses elementary methods to develop the formulas for the number of direct-sum decompositions that are the q-analogs of the formulas for: (1) the number of set partitions with a given number partition signature; (2) the number of set partitions of an n-element set with m blocks (the Stirling numbers of the second kind); and (3) for the total number of set partitions of an n-element set (the Bell numbers). The paper also develops the formulas to enumerate: (4) the number of direct-sum decompositions in an n-dimensional vector space over GF (q) with m blocks and where any given nonzero vector is in one of the blocks, and (5) the number of direct-sum decompositions in an n-dimensional vector space over GF(q) where any given nonzero vector is in one of the blocks.

In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate the thermodynamics of protein folding. In this paper, we present new characterizations of the Hamiltonian cycles in labeled triangular grid graphs, which are graphs constructed from rectangular grids by adding a diagonal to each cell. By using these characterizations and implementing the computational method outlined here, we confirm the existing data, and obtain some new results that have not been published. A new interpretation of Catalan numbers is also included.

Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors have shown that Patience Sorting provides an algorithmic description for permutations avoiding the barred (generalized) permutation pattern 3−1−42. Motivated by this and a recently formulated geometric form for Patience Sorting in terms of certain intersecting lattice paths, we study the related themes of restricted input and avoidance of similar barred permutation patterns. One such result is to characterize those permutations for which Patience Sorting is an invertible algorithm as the set of permutations simultaneously avoiding the barred patterns 3−1−42 and 3−1−24. We then enumerate this avoidance set, which involves convolved Fibonacci numbers.

Abstract In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate the thermodynamics of protein folding. In this paper, we present new characterizations of the Hamiltonian cycles in labeled triangular grid graphs, which are graphs constructed from rectangular grids by adding a diagonal to each cell. By using these characterizations and implementing the computational method outlined here, we confirm the existing data, and obtain some new results that have not been published. A new interpretation of Catalan numbers is also included.

Given a universe of discourse U , a multiset can be thought of as a function M from U to the natural numbers N. In this paper, we define a hybrid set to be any function from the universe U to the integers Z. These sets are called hybrid since they contain elements with either a positive or negative multiplicity. Our goal is to use these hybrid sets as if they were multisets in order to adequately generalize certain combinatorial facts which are true classically only for nonnegative integers. However, the definition above does not tell us much about these hybrid sets; we must define operations on them which provide us with the needed combinatorial structure.

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for $m\leq10$. The computational data we gathered suggests that $h^A_m(n)\sim h^B_m(n)$ when $m$ is even.

This paper is a continuation of the study of partially ordered patterns (POPs) introduced recently. We provide a general approach to code combinatorial objects using (POP-)restricted permutations. We give several examples of relations between permutations restricted by POPs and other combinatorial structures, such as labeled graphs, walks, binary vectors, and others. Also, we show how restricted permutations are related to Cartesian products of certain objects.

We use the conceptual idea of "maps on orbifolds" and the theory of the non-Euclidian crystallographic groups (NEC groups) to enumerate rooted and unrooted maps (both sensed and unsensed) on surfaces regardless of genus. As a consequence we deduce a formula for the number of chiral pairs of maps. The enumeration principle used in this paper is due to , it counts the number of conjugacy classes of subgroups in NEC groups which are in one-two-one correspondence with unrooted (sensed or unsensed) maps.

This paper is a continuation of the study of partially ordered generalized patterns (POGPs) considered in . We provide two general approaches: one to obtain connections between restricted permutations and other combinatorial structures, and one to treat the avoidance problems for words in case of segmented patterns. The former approach is related to coding combinatorial objects in terms of restricted permutations. We provide several examples of relations of our objects to other combinatorial structures, such as labeled graphs, walks, binary vectors, and others. Also, we show how restricted permutations are related to Cartesian products of certain objects.

We give a recursive formula for the Möbius function of an interval [σ, π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1, 2, . . . , k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. A consequence of the formula is that the Möbius function of such an interval [σ, π] is bounded by the number of occurrences of σ as a pattern in π. We also show that for any separable permutation π the Möbius function of (1, π) is either 0, 1 or −1.

It is shown that the sequence of the generalized Bell polynomials Sn(x) is convex under some restrictions of the parameters involved. A kind of recurrence relation for Sn(x) is established, and some numbers related to the generalized Bell numbers and their properties are investigated.