Combinatorial Sums Research Articles

Several computing formulas for combinatorial sums

In this paper, by Riordan array several computing formulas for the combinatorial sums are given.

The Bott Formula for Toric Varieties

The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $\Omega_{\P}^p(D)$ of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for ${\O_X}(D)$. We also introduce a new combinatorial object, the so-called p-th Hilbert-Erhart polynomial, which generalizes the usual notion and behaves similar. Namely, there exists a generalization of the inversion law for a usual Hilbert-Erhart polynomial. Some applications of the Bott formula are discussed.

Quantum graphs: a model for quantum chaos

We study the statistical properties of the scattering matrix associated with generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spectral form factor of the scattering matrix we have recently derived an exact combinatorial expression. It is based on a sum over families of periodic orbits which so far could only be performed in special graphs. Here we present a simple algorithm implementing this summation for any graph. Our results are in excellent agreement with direct numerical simulations for various graphs. Moreover, we extend our previous notion of an ensemble of graphs by considering ensemble averages over random boundary conditions imposed at the vertices. We show numerically that the corresponding form factor follows the predictions of random-matrix theory when the number of vertices is large – even when all bond lengths are degenerate. The corresponding combinatorial sum has a structure similar to the one obtained previously by performing an energy average under the assumption of incommensurate bond lengths.

Combinatorial Sums and Series Involving Inverses of Binomial Coefficients

Combinatorial Sums and Series Involving Inverses of Binomial Coefficients

Asymptotics of multinomial sums and identities between multi-integrals

We calculate the asymptotics of combinatorial sums ∑αf(α)(αn)β, whereα = (α1, …,αh) withαi =αj for certaini, j. Hereh is fixed and theαi’s are natural numbers. This implies the asymptotics of the correspondingSn-character degrees ∑λf(λ)dλβ. For certain sequences ofSn characters which involve Young’s rule, the latter asymptotics were obtained earlier [1] by a different method. Equating the two asymptotics, we obtain equations between multi-integrals which involve Gaussian measures. Special cases here give certain extensions of the Mehta integral [5], [6].

Sections and Lacunary Sums of Linearly Recurrent Sequences

Linear recurrent sequences have been extensively studied for a long time because of their importance in combinatorics, difference equations, Ž w x. number theory, algebra, etc. see Refs. 1, 4, 5, 7 . Many combinatorial objects, like combinatorial sums, are linearly recurrent sequences as funcŽ w x. tions of one of their parameters see 2, 6 . An important example is the w x class of lacunary sums of binomial coefficients. Howard and Witt 3 studied recently some aspects of such combinatorial sums using the method of multisection of series. In this paper we study some aspects of the linearly recurrent sequences Ž . LRS related to sums of binomial coefficients. In particular we study certain linear operators on the vector space of LRS that include sections Ž . Ž . Ž . and sums. A section of a sequence F k is a sequence G k s F rk q s , Ž . where r and s are fixed integers. The sum of F k is the sequence Ž . my 1 Ž . SF m s Ý F k , and a lacunary sum of F is the sum of a section ks0 of F. In our development we use the basis of the space of LRS that consists of

Singularity analysis and asymptotics of Bernoulli sums

The asymptotic analysis of a class of binomial sums that arise in information theory can be performed in a simple way by means of singularity analysis of generating functions. The method developed extends the range of applicability of singularity analysis techniques to combinatorial sums involving transcendental elements like logarithms or fractional powers.

Asymptotic Estimates Using Probability

Probabilistic methods are used to asymptotically estimate certain combinatorial sums. When applied to Young-derived sequences of characters of symmetric groups, these asymptotics yield interesting integration formulas.

Left-inversion of combinatorial sums

The inversion of combinatorial sums is a fundamental problem in algebraic combinatorics. Some combinatorial sums, such as a n = Σ k d n,kb k , cannot be inverted in terms of the orthogonality relation because the infinite, lower triangular array P = { d n,k }'s diagonal elements are equal to zero (except d 0,0). Despite this, we can find a left-inverse ̄P such that PP̄ = I and therefore are able to left-invert the original combinatorial sum, and thus obtain b n = Σ k d̄ n,ka k .

Analytic structure of the one-dimensional random-bond Ising model

The quenched average of the one-dimensional random-bond (+or-J) Ising model in a magnetic field is studied using a technique based on the transformation of a product of random 2*2 matrices to an iterated conformal map. This approach allows for the underlying analytic structure of the random matrix product to appear as a natural consequence of the conformal mapping of the extended complex plane. The quenched average of the characteristic exponent (i.e. the free energy) is obtained by averaging over a specified probability distribution of nearest-neighbour coupling. The evaluation of the quenched average of the characteristic exponent is performed within the isoentropic approximation. The isoentropic approximation treats all bond configurations with the same number of antiferromagnetic domains having the same entropy and enables the enumeration of equivalent configurations. Combinatorial methods can then be used to express the average over all realizations of the random matrix product as a combinatorial sum. The combinatorial sum is evaluated using resummation methods and an explicit expression for the quenched average of the characteristic exponent is thereby obtained for the case of the random-bond Ising model. The explicit expression for the free energy is dependent on constants which are calculated numerically.

Riordan arrays and combinatorial sums

The concept of a Riordan array is used in a constructive way to find the generating function of many combinatorial sums. The generating function can then help us to obtain the closed form of the sum or its asymptotic value. Some examples of sums involving binomial coefficients and Stirling numbers are examined, together with an application of Riordan arrays to some walk problems.

Reciprocal Sums over Partitions and Compositions

The authors obtain precise asymptotic estimates for certain combinatorial sums over products of reciprocals of the summands in the partition or composition of a natural number n. These estimates are applied to determine the mean value of a certain arithmetical function over the polynomial ring $\mathbb{F}_q [ X ]$, where $\mathbb{F}_q $ is the finite field with q elements.

A unified treatment of a class of combinatorial sums

Here introduced is a class of combinatorial sums that can be treated by means of an embedding and inversion technique. Some classic identities and novel ones are demonstrated to be members of the class defined. Solution of Liskovets' problems is reconsidered, and an additional class of identities is formulated.

Problems of combinatorial optimization, statistical sums and representations of the full linear group

Problems of combinatorial optimization, statistical sums and representations of the full linear group

Big Hankel operators of higher weight

Known results concerning the smoothness and boundedness of «big» Hankel operators (Hankel operators in the sense of Axler) are generalized to the case of higher weight (in the sense of representation theory). The key result is a certain estimate for thes-numbers of a particular such operator, involving a combinatorial sum.

Asymptotics of Combinatorial Sums and the Central Limit Theorem

The aim of this paper is to generalize the results of Regev [Adv. in Math., 41 (1981), pp. 115–136] and to simplify the proofs by deducing them from the Central Limit Theorem of probability theory. The generalized results also yield a method for calculating certain multi-integrals, some of which seem highly nontrivial.

Height probabilities in solid-on-solid models. I

The height probabilities for some infinite sequences of solid-on-solid models are given in terms of combinatorial sums for the large but finite lattice. The authors transform a class of these sums to a form suitable for taking the infinite lattice limit.

Register Allocation for Unary–Binary Trees

We study the number of registers required for evaluating arithmetic expressions formed with any set of unary and binary operators. Our approach consists in a singularity analysis of intervening generating functions combined with a use of (complex) Mellin inversion. We illustrate it first by rederiving the known results about binary trees and then extend it to the fully general case of unary–binary trees. The method used, as mentioned in the conclusion, is applicable to a wide class of combinatorial sums.

Combinatorial sums [formula omitted] associated with Chebyshev polynomials

Combinatorial sums [formula omitted] associated with Chebyshev polynomials

Combinatorial sums, identities and trace identities of the 2 × 2 matrices

An asymptotic formula, involving integrals, is given for certain combinatorial sums. By evaluating a multi-integral it is then found that as n → ∞, the codimensions c n ( F 2) and the trace codimensions t n ( F 2) of F 2, the 2 × 2 matrices, are asymptotically equal: c n(F 2) ⋍ t n(F 2) .