On the calculation of generalized binomial coefficients

The generalized binomial coefficients (_κ^λ) are defined by

$\text{Cλ(I}{}_{\mathrm{m}}\text{+ R)}\text{Cλ(I}{}_{\mathrm{m}}\text{)}\text{=}\text{Σ}\text{k=0}\text{l}\text{Σ}\text{κ}\text{(}\text{λ}\text{κ}\text{)}\text{C}{}_{\mathrm{\kappa }}\text{(R)}\text{C}{}_{\mathrm{\kappa }}\text{(I}{}_{\mathrm{m}}\text{)}$

, where the C_k(R) are the zonal polynomials of the m × m matrix R. In this paper some simple expressions are derived which allow straightforward calculation of a large number of these coefficients.     

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

On a combinatorial problem related to permanents

Some numerical results will be presented concerning the largest and smallest values of square permanents containing the integers 1, 2, 3, ...,n ² n=1(1)8.Dedicated to Peter Naur on the occasion of his 60th birthday     

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n₁,…,nm, nm+1=n₁, and any nonnegative integer r, the expression

     

Arithmetic-Geometric Mean determinantal identity

In this paper, we give a generalization of a determinantal identity posed by Charles R. Johnson about minors of a Toeplitz matrix satisfying a specific matrix identity. These minors are those appear in the Dodgson’s condensation formula.     

On minors of the compound matrix of a Laplacian

Let L   be an n×n$n\times n$ matrix with zero row and column sums, n?3$n?3$. We obtain a formula for any minor of the (n−2)$(n-2)$-th compound of L. An application to counting spanning trees extending a given forest is given.     

Modular periodicity of binomial coefficients

We prove that if the signed binomial coefficient viewed modulo p is a periodic function of i with period h in the range 0?i?k, then k+1 is a power of p, provided h is not too large compared to k. (In particular, 2h?k suffices). As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H<G, and such that 1-α∈G for all α∈G?H, then G∪{0} is a subfield.     

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we have

     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

Linear recurrence relations for binomial coefficients modulo a prime

We investigate when the sequence of binomial coefficients modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0?i?k. In particular, we prove that this cannot occur if 2h?k<p−h. This hypothesis can be weakened to 2h?k<p if we assume, in addition, that the characteristic polynomial of the relation does not have −1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.     

On Some Properties of Permutation Tableaux

We consider the relations between various permutation statistics and properties of permutation tableaux. We answer some of the open problems of Steingrímsson and Williams [8], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. Research supported in part by the NSA Young Investigator Grant H98230-06-1-0037.     

On product of companion matrices

This paper describes an explicit combinatorial formula for the product of companion matrices. The method relies on the connections between matrix algebra and associated combinatorial structures to enumerate the paths in an unweighted digraph. As an application, we obtain bases for the solution space of the linear difference equation with variable coefficients.     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(M_n(D)) and determine the diameters of some induced subgraphs of Γ(M_n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M_n(D). For every field F, it is shown that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 6. We conjecture that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.     

On the number of homogeneous subgraphs of a graph

Let us defineG(n) to be the maximum numberm such that every graph onn vertices contains at leastm homogeneous (i.e. complete or independent) subgraphs. Our main result is exp (0.7214 log² n) ≧G(n) ≧ exp (0.2275 log² n), the main tool is a Ramsey—Turán type theorem. We formulate a conjecture what supports Thomason’s conjecture R(k, k)^1/k = 2.     

On congruences related to central binomial coefficients

It is known that