On the Pólya Enumeration Theorem

Enumeration of lozenge tilings of halved hexagons with a boundary defect

We generalize a special case of a theorem of Proctor on the enumeration of lozenge tilings of a hexagon with a maximal staircase removed, using Kuo's graphical condensation method. Additionally, we prove a formula for a weighted version of the given region. The result also extends work of Ciucu and Fischer. By applying the factorization theorem of Ciucu, we are also able to generalize a special case of MacMahon's boxed plane partition formula.

The Extension of the Desargues Theorem, the Converse, Symmetry and Enumeration

This work is, in part, a generalization of the article by A.A. Bruen ,T.C Bruen and J.M.McQuillan on Desargues Theorem in arXiv:2007.09175[mathCO]July 17,2020. We prove the extension of Desargues theorem in all dimensions, using 4 different arguments.We also show the converse theorem. It is shown that the Desargues configuration in projective n-space, for all n at least 2, corresponds to an arc in a projective space of dimension n+1 containing the n-space. Thus, in principle, one can enumerate the number of Desargues configurations in n dimensions when the underlying field is finite. The Desar…

Numerical Algorithm for Pólya Enumeration Theorem

Although the Pólya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of unique colorings of a finite set under the action of a finite group.

Weighted enumeration of spanning subgraphs with degree constraints

The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger multivariate version of this theorem. We provide a general method by which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph $G$. These polynomials are multivariate generating functions for spanning subgraphs of $G$ with certain weights and constraints imposed, and the theorems specify regions in which these po…

Enumeration of spanning subgraphs with degree constraints

For a finite undirected multigraph G=(V,E) and functions f,g:V-->\NN, let N_f^g(G,j) denote the number of (f,g)-factors of G with exactly j edges. The Heilmann-Lieb Theorem implies that \sum_j N_0^1(G,j) t^j is a polynomial with only real (negative) zeros, and hence that the sequence {N_0^1(G,j)} is strictly logarithmically concave. Separate generalizations of this theorem were obtained by Ruelle and by the author. We unify, simplify, and generalize these results by means of the Grace-Szegö-Walsh Coincidence Theorem.

通过有序树自然枚举重新诠释中层定理

Reinterpreting the Middle-Levels Theorem via Natural Enumeration of Ordered Trees

本文通过构造中层图的二面体商图，将哈密顿圈存在性证明重新解释为对阶为 k+1、大小为 k 的有序有根树的自然枚举，从而为中层定理提供了新的组合视角。

A $q$-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary

MacMahon proved a simple product formula for the generating function of plane partitions fitting in a given box. The theorem implies a $q$-enumeration of lozenge tilings of a semi-regular hexagon on the triangular lattice. In this paper we generalize MacMahon's classical theorem by $q$-enumerating lozenge tilings of a new family of hexagons with four adjacent triangles removed from their boundary.

A shuffling theorem for lozenge tilings of doubly-dented hexagons

MacMahon's theorem on plane partitions yields a simple product formula for tiling number of a hexagon, and Cohn, Larsen and Propp's theorem provides an explicit enumeration for tilings of a dented semihexagon via semi-strict Gelfand--Tsetlin patterns. In this paper, we prove a natural hybrid of the two theorems for hexagons with an arbitrary set of unit triangles removed along a horizontal axis. In particular, we show that the `shuffling' of removed unit triangles only changes the tiling number of the region by a simple multiplicative factor. Our main result generalizes a number of known enume…

具有有限最大度数的图中 clique 的高效枚举

Efficient Enumeration of Cliques in Graphs with Bounded Maximum Degree

本文提出了一种高效方法用于枚举具有有限最大度数的图中的 clique，并通过新证明改进了 Cutler-Radcliffe 定理和 Kahn-Zhao 定理。

Structure and enumeration theorems for hereditary properties in finite relational languages

Given a finite relational language $\calL$, a hereditary $\calL$-property is a class of finite $\calL$-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary $\calL$-properties, where $\calL$ is any finite relational language with maximum arity at le…

Different classes of binary necklaces and a combinatorial method for their enumerations

In this paper we investigate enumeration of some classes of $n$-character strings and binary necklaces. Recall that binary necklaces are necklaces in two colors with length $n$. We prove three results (Theorems 1, 1' and 2) concerning the numbers of three classes of $j$-character strings (closely related to some classes of binary necklaces or Lyndon words). Using these results, we deduce Moreau's necklace-counting function for binary aperiodic necklaces of length $k$ \cite{mo} (Theorem 3), and we prove the binary case of MacMahon's formula from 1892 \cite{ma} (also called Witt's formula) for t…

A bijective enumeration of $3$-strip tableaux

Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical André's theorem for the number of up-down permutations. They asked for a bijective proof for the enumeration of $3$-strip tableaux. In this paper we will provide such a bijective proof. First we count the $3$-strip tableaux by decomposition. Secondly we will apply this "decomposition" idea on the up-down permutations and down-up permutations to enumerate the $3$-strip tableaux bijectively.

A Generalization of Polya's Enumeration Theorem or the Secret Life of Certain Index Sets

Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $χ$ of $W$, the polynomial functor $F_χ$ corresponding via S-MT to the induced monomial representation $U_χ= ind_W^{S_d}(χ)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_χ)$ is the weighted inventory of some set $J(χ)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(χ) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_χ…

Enumeration of Hybrid Domino-Lozenge Tilings

We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglas' Theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.

Enumeration and Random Generation of Unlabeled Classes of Graphs: A Practical Study of Cycle Pointing and the Dissymmetry Theorem

Our work studies the enumeration and random generation of unlabeled combinatorial classes of unrooted graphs. While the technique of vertex pointing provides a straightforward procedure for analyzing a labeled class of unrooted graphs by first studying its rooted counterpart, the existence of nontrivial symmetries in the unlabeled case causes this technique to break down. Instead, techniques such as the dissymmetry theorem (of Otter) and cycle pointing (of Bodirsky et al.) have emerged in the unlabeled case, with the former providing an enumeration of the class and the latter providing both an…

The enumeration of planar graphs via Wick's theorem

A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type p…

Automatic Enumeration of Generalized Menage Numbers

I describe an empirical-yet-rigorous, algorithm, based on Riordan's rook polynomials and the so-called C-finite ansatz, fully implemented in the accompanying Maple package (http://www.math.rutgers.edu/~zeilberg/tokhniot/MENAGES ), MENAGES, that reproduces in a few seconds, rigorously-proved enumeration theorems on permutations with restricted positions, previously proved by quite a few illustrious human mathematicians, and that can go far beyond any human attempts.

A non-backtracking Polya's theorem

Pólya's random walk theorem states that a random walk on a $d$-dimensional grid is recurrent for $d=1,2$ and transient for $d\ge3$. We prove a version of Pólya's random walk theorem for non-backtracking random walks. Namely, we prove that a non-backtracking random walk on a $d$-dimensional grid is recurrent for $d=2$ and transient for $d=1$, $d\ge3$. Along the way, we prove several useful general facts about non-backtracking random walks on graphs. In addition, our proof includes an exact enumeration of the number of closed non-backtracking random walks on an infinite 2-dimensional grid. This …

Enumeration of Graded (3+1)-Avoiding Posets

The notion of (3+1)-avoidance has shown up in many places in enumerative combinatorics. The natural goal of enumeration of all (3+1)-avoiding posets remains open. In this paper, we enumerate graded (3+1)-avoiding posets for both reasonable definitions of the word "graded." Our proof consists of a number of structural theorems followed by some generating function magic. We also provide asymptotics for the growth rate of the number of graded (3 + 1)-avoiding posets.

Enumeration of Linear Transformation Shift Registers

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.

Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems

In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions, so the natural question for the number of representations arises. In this paper, we provide a general asymptotic formula for the number of such multi-base representations of a positive integer $n$. Moreover, we prove central limit theorems …

无标号三次平面图的渐近计数

Asymptotic enumeration of unlabelled cubic planar graphs

本文结合生成函数方法、计算界和概率局部大偏差定理，精确确定了n个顶点的无标号三次平面图的渐近数量。

伪线图和伪环图的魔术标记计数

Magic labelling enumeration on pseudo-line graphs and pseudo-cycle graphs

本文研究伪线图和伪环图的魔术标记计数及其生成函数，扩展了Bóna等人的早期研究成果。

Polya by Examples

Polya Enumeration Theorem is one of the most useful tools dealing with the enumeration of patterns that are symmetric in some ways. What follows is a procedure for obtaining the results of Polya Theorem directly, bypassing the usual preliminaries.

Enumeration of lozenge tilings of a hexagon with a maximal staircase and a unit triangle removed

Proctor proved a formula for the number of lozenge tilings of a hexagon with side-lengths $a,b,c,a,b,c$ after removing a "maximal staircase." Ciucu then presented a weighted version of Proctor's result. Here we present weighted and unweighted formulas for a similar region which has an additional unit triangle removed. We use Kuo's graphical condensation method to prove the results. By applying the factorization theorem of Ciucu, we obtain a formula for the number of lozenge tilings of a hexagon with three holes on consecutive edges.

Refined enumeration of $k$-plane trees and $k$-noncrossing trees

A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that …

双循环有向图的计数

Enumeration of dicirculant digraphs

本文利用 Pólya 计数定理，对阶数 $4p$（$p$ 为素数）的双循环有向图及连通情形进行同构类计数，并给出各出度 $k$ 下的数量。

A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a …

平面(3,6)-三角剖分的计数与奇完全数的形式

The enumeration of plane (3,6)-triangulations and the form of odd perfect numbers

该研究计数了顶点度数为3或6的3-连通平面三角剖分族中特定度数的非同构剖分数目，并借助奇数参数下的计数公式强化了关于奇完全数形式的欧拉定理。

Enumeration of symmetric centered rhombus tilings of a hexagon

A rhombus tiling of a hexagon is said to be centered if it contains the central lozenge. We compute the number of vertically symmetric rhombus tilings of a hexagon with side lengths $a, b, a, a, b, a$ which are centered. When $a$ is odd and $b$ is even, this shows that the probability that a random vertically symmetric rhombus tiling of a $a, b, a, a, b, a$ hexagon is centered is exactly the same as the probability that a random rhombus tiling of a $a, b, a, a, b, a$ hexagon is centered. This also leads to a factorization theorem for the number of all rhombus tilings of a hexagon which are cen…

Enumeration of lozenge tilings of a hexagon with a shamrock missing on the symmetry axis

In their paper about a dual of MacMahon's classical theorem on plane partitions, Ciucu and Krattenthaler proved a closed form product formula for the tiling number of a hexagon with a "shamrock", a union of four adjacent triangles, removed in the center (Proc. Natl. Acad. Sci. USA 2013). Lai later presented a $q$-enumeration for lozenge tilings of a hexagon with a shamrock removed from the boundary (European J. Combin. 2017). It appears that the above are the only two positions of the shamrock hole that yield nice tiling enumerations. In this paper, we show that in the case of symmetric hexago…

Face enumeration on simplicial complexes

Let $M$ be a closed triangulable manifold, and let $Δ$ be a triangulation of $M$. What is the smallest number of vertices that $Δ$ can have? How big or small can the number of edges of $Δ$ be as a function of the number of vertices? More generally, what are the possible face numbers ($f$-numbers, for short) that $Δ$ can have? In other words, what restrictions does the topology of $M$ place on the possible $f$-numbers of triangulations of $M$? To make things even more interesting, we can add some combinatorial conditions on the triangulations we are considering (e.g., flagness, balancedness, et…

8p阶非阿贝尔群上Cayley图的计数

Enumeration of Cayley graphs over a nonabelian group of order $8p$

针对8p阶非阿贝尔群$T_{8p}$，利用波利亚计数定理给出互不同构Cayley图及连通Cayley图的计数公式，并列出3≤p≤13时的精确数量。

Generalized Monotone Triangles: an extended Combinatorial Reciprocity Theorem

In a recent work, the combinatorial interpretation of the polynomial alpha(n;k1,k2,...,kn) counting the number of Monotone Triangles with bottom row k1 < k2 < ... < kn was extended to weakly decreasing sequences k1 >= k2 >= ... >= kn. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles - a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of alpha(n;k1,k2,...,kn) at a…

关于 $l$-超地图的计数

On enumeration of $l$-hypermaps

本文基于特定定理与矩阵预解法，推导出 $l$-超地图计数的 $k$-点生成级数显式公式，并推广了 $l=2$ 情形及 Dubrovin 定理。

Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in NCP_W as a generalised Fuss-Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the …

A formula for the doubly-refined enumeration of alternating sign matrices

Zeilberger proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey, for the number of alternating sign matrices with given top row. Stroganov proved an explicit formula for the number of alternating sign matrices with given top and bottom rows. Fischer and Romik considered a different kind of "doubly-refined enumeration" where one counts alternating sign matrices with given top two rows, and obtained partial results on this enumeration. In this paper we continue the study of the doubly-refined enumeration with respect to…

Enumeration and randomized constructions of hypertrees

Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog of Cayley's formula for the number of $n$-vertex trees. He enumerated $d$-dimensional hypertrees weighted by the squared size of their $(d-1)$-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of $d$-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of $d$-hypertrees. In addition, we study a random $1$-out model of $d$-complexes where every $(d-1)$-dimensional face selects a random $d$-face cont…

Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs

A \emph{Golomb ruler} is a sequence of distinct integers (the \emph{markings} of the ruler) whose pairwise differences are distinct. Golomb rulers can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for \emph{optimal} Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study [g_m(t) := # {\x \in \Z^{m+1} : \, 0 = x_0 < x_1 < ... < x_{m-1} < x_m = t, \text{all} x_j - x_k \text{distinct}},] the number of Golomb rulers w…

Pólya enumeration theorems in algebraic geometry

We generalize a formula due to Macdonald that relates the singular Betti numbers of $X^{n}/G$ to those of $X$, where $X$ is a compact manifold and $G$ is any subgroup of the symmetric group $S_{n}$ acting on $X^{n}$ by permuting coordinates. Our result is completely axiomatic: in a general setting, given an endomorphism on the cohomology $H^{\bullet}(X)$, it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on $H^{\bullet}(X^{n})^{G}$ to that of the given endomorphism on $H^{\bullet}(X)$ in the presence of the Künneth formula with respect to a cup product. …

Enumeration of points, lines, planes, etc

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points $E$ in a projective space determines at least $|E|$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of a $d$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E…

Linear algebraic techniques for spanning tree enumeration

Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally or algebraically taxing. We show how two well-known results from linear algebra, the Matrix Determinant Lemma and the Schur complement, can be used to elegantly count the spanning trees in several significant families of graphs.

具有给定顶点数、自环和重边的树状多重图的枚举

Enumeration of Tree-like Multigraphs with a Given Number of Vertices, Self-loops and Multiple Edges

提出统一动态规划框架，利用规范有根表示和递归分解，精确枚举具有给定顶点数、自环和重边的非同构树状多重图，并提供解析界限和递推关系。

Enumeration of Random Walk Positions in $L_1$-norm ball in $\mathbb{Z}^d$

In this paper, we mainly concerned about deriving the general formula to count the possible positions of $n$ step random walk in $\mathbb{Z}^d$ with unit length in each step, which we denoted as $|P_n^{d}|$. For our results, we firstly propose a recurrence relation of the counting formula: $|P_n^{d+1}| = |P_n^d| + 2\sum_{k=0}^{n-1} |P_k^d|$. Next, we propose two methods in deriving the explicit formula of $|P_n^{d}|$ using generating functions and Faulhaber's formula. Finally, we reached our main theorem in the matrix representation of our formula.

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration …