Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are related to certain configurations of vicious random walkers and to the enumeration of plane partitions. The combinatorial meaning of the average of the characteristic polynomial of random Hermitian and Wishart matrices is also investigated, and consequently several simple universality results are derived.
We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G_2 holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U(1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside Lie groups. We also point out certain cases …
We deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.
I present a recursive formula for calculating the number of abelian squares of length $n+n$ over an alphabet of size $d$. The presented formula is similar to a previously known formula but has substantially lower complexity when $d\gg n$.
Koroljuk gave a summation formula for counting the number of lattice paths from $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that stay strictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. In this paper we obtain an explicit formula for the number of lattice paths from $(a,b)$ to $(m,n)$ above the diagonal $y=kx-r$, where $r$ is a rational number. Our result slightly generalizes Koroljuk's formula, while the former can be essentially derived from the latter. However, our proof uses a recurrence with respect to the starting points, and hereby presents a new a…
Harary and Palmer announced an enumeration problem of labelled self-complementary graphs at the end of their book (Graphical Enumeration, Academic Press, New York and London, 1973). This paper resolves this problem. A method for solving this problem leads to the derivation of following formulas: (a) A formula on the number of labelled graphs with the given order of automorphism groups of those graphs. (b) A formula on the number of unlabelled graphs with the given order of automorphism groups of those graphs. (c) A formula on the number of labelled self-complementary graphs with the given orde…
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some results by Knuth and by Bousquet-Mélou and Chapuy about embedded trees. We also give a new proof of the multivariate Lagrange inversion formula. Our strategy for counting trees is to exploit symmetries of refined enumeration formulas: proving these symmetries is easy, and once the symmetries are proved the formulas follow effortlessly. We also adapt this strate…
Fondazione Bruno Kessler（布鲁诺·凯斯勒基金会）；University of Trento（特伦托大学）
Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed-forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, …
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, and used it to compute $p_n$, the number of parabolic double cosets in $S_n$, for $n\leq13$. In this paper, we derive a new formula for $p_n$ and an efficient polynomial time algorithm for evaluating this formula. We use these results to compute $p_n$ for $n\leq5000$ and to prove an asymptotic formula for $p_n$ that was conjectured by Billey et al.
This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous polynomials of a given degree is derived. This admits an interpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane.
Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch, Ramgoolam, and Wen [Nuclear Phys.\,B {\bf 870} (2013), 530--581] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an expl…
In this paper, we begin with the Lehman-Walsh formula counting one-face maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is in the form of a convolution of the Stirling numbers of the first kind which immediately implies a formula for the generating function $A_n(x)=\sum_{g\geq 0}A(n,g)x^{n+1-2g}$ other than the well-known Harer-Zagier formula. By reformulating our expression for $A_n(x)$ in terms of the backward shift operator $E: f(x)\rightarrow f(x-1)$ and proving a property satisfied b…
We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientat…
In the present article we shown a formula to compute the number of all matrices over the finite field $F$ whit prescribed eigenvalues. Using this formula we obtain one inequality for the number of $(k+1)$-potent elements over finite rings.
This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some known enumerations are recovered and several new ones are provided.
We consider the problem of counting $k$-cliques in $s$-uniform Erdos-Renyi hypergraphs $G(n,c,s)$ with edge density $c$, and show that its fine-grained average-case complexity can be based on its worst-case complexity. We prove the following: 1. Dense Erdos-Renyi graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized ETH, it takes $n^{Ω(k)}$ time to count $k$-cliques in $G(n,c,s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi graphs and hypergraphs: When $c = Θ(n^…
We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman--Stanley polytope of an n-tuple.
We consider lattice walks in the plane starting at the origin, remaining in the first quadrant and made of West, South and North-East steps. In 1965, Germain Kreweras discovered a remarkably simple formula giving the number of these walks (with prescribed length and endpoint). Kreweras' proof was very involved and several alternative derivations have been proposed since then. But the elegant simplicity of the counting formula remained unexplained. We give the first purely combinatorial explanation of this formula. Our approach is based on a bijection between Kreweras walks and triangulations w…
An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to $n$ as $n$ goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to $n$ and using exactly $k$ multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers $n$, we compare the lengths o…
A formula for the number of toroidal m x n binary arrays, allowing rotation of the rows and/or the columns but not reflection, is known. Here we find a formula for the number of toroidal m x n binary arrays, allowing rotation and/or reflection of the rows and/or the columns.
In a recent work I developed a formula for efficiently calculating the number of abelian squares of length $t+t$ over an alphabet of size $d$, where $d$ may be very large. Here I show how the expressiveness of a certain class of parameterized quantum circuits can be reduced to the problem of counting abelian squares over a large alphabet, and use the recently developed formula to efficiently calculate this quantity.
This paper deals with the distribution of descent number in standard Young tableaux of certain shapes. A simple explicit formula is presented for the number of tableaux of any shape with two rows, with any specified number of descents. For shapes of three rows with one cell in the third row recursive formulas are given, and are solved in certain cases.
The Baldoni--Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function fundamental in representation theory. On the other hand the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these f…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$
A sharp asymptotic formula for the number of strongly connected digraphs on $n$ labelled vertices with $m$ arcs, under a condition $m-n\to\infty$, $m=O(n)$, is obtained; this solves a problem posed by Wright back in $1977$. Our formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on $n$ vertices with $m$ edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Ni…
Cayley's formula states that there are $n^{n-2}$ spanning trees in the complete graph on $n$ vertices; it has been proved in more than a dozen different ways over its 150 year history. The complete graphs are a special case of threshold graphs, and using Merris' Theorem and the Matrix Tree Theorem, there is a strikingly simple formula for counting the number of spanning trees in a threshold graph on $n$ vertices; it is simply the product, over $i=2,3, ...,n-1$, of the number of vertices of degree at least $i$. In this manuscript, we provide a direct combinatorial proof for this formula which d…
In this paper we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surfaces using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms.
We prove that there is an finite number of friezes in type D_n, and we provide a formula to count them. As a corollary, we obtain formulas to count the number of friezes in types B_n, C_n and G_2. We conjecture finiteness (and precise numbers) for other Dynkin types.
An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as…
For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $τ_G(M)$ is obtained, where $τ_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanni…
We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we answer some questions by M. T$\ddot{a}$rn$\ddot{a}$uceanu in \cite{MT} and L. T$\dot{\acute{o}}$th in \cite{LT}. We also use other methods such as the method of fundamental group lattices introduced in \cite{MT} to derive a similar counting function in a special case of arbitrary rank finite abelian p-groups.
We state and prove some counting formulas relating to cliques in the distant graphs of projective lines over finite rings. As a preliminary to this, we prove a decomposition theorem for the graphs in terms of the direct-product decomposition of their rings.
After defining convex near-polygons, a formula enumerating the number of triangulations of such configurations is derived in terms of edge-polynomials. The paper describes also a transfer-matrix approach for computing quantities related to triangulations.
Define $QC(n)$ to be the number of quasiplatonic topological actions of the cyclic group $C_n$ on surfaces of genus at least two. We use formulas of Benim and Wootton to give an explicit formula for $QC(n)$. In addition, we relate the number of quasiplatonic topological actions of $C_n$ to the number of regular dessins d'enfants having $C_n$ as a group of automorphisms.
This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface is the projective plane or the product of two projective lines then the invariants under consideration coincide with the Gromov-Witten invariants. The formula gives a new count even in these cases, where other computational technique is available.
