q-Analogs of some congruences involving Catalan numbers

We present some variations on the Greene–Krammer?s identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.     

<Emphasis Type="Italic">q</Emphasis>-Fibonacci Polynomials and the Rogers-Ramanujan Identities

We derive some formulas for the Carlitz q-Fibonacci polynomials F_n(t) which reduce to the finite version of the Rogers-Ramanujan identities obtained by I. Schur for t = 1. Our starting point is a representation of the q-Fibonacci polynomials as the weight of certain lattice paths in contained in a strip along the x-axis. We give an elementary combinatorial proof by using only the principle of inclusion-exclusion and some standard facts from q-analysis.     

A p, q-Analogue of the Generalized Derangement Numbers

In this paper, we study the numbers D _n,k which are defined as the number of permutations σ of the symmetric group S _n such that σ has no cycles of length j for j ≤ k. In the case k = 1, D _n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D _n,k generalized derangement numbers. Garsia and Remmel [4] defined some natural q-analogues of D _n,1, denoted by D _n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues D _n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In [4], Garsia and Remmel also suggested an approach to define q-analogues of the numbers D _n,k . In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2. Partially supported by NSF grant DMS 0400507.     

Bit strings withoutq-separation

LetB(n, q) denote the number of bit strings of lengthn withoutq-separation. In a bit string withoutq-separation no two 1's are separated by exactlyq – 1 bits.B(n, q) is known to be expressible in terms of a product of powers of Fibonacci numbers. Two new and independent proofs are given. The first proof is by combinatorial enumeration, while the second proof is inductive and expressesB(n, q) in terms of a recurrence relation.     

Conjectured statistics for the q,t-Catalan numbers

We introduce the distribution function F_n(q,t) of a pair of statistics on Catalan words and conjecture F_n(q,t) equals Garsia and Haiman's q,t-Catalan sequence C_n(q,t), which they defined as a sum of rational functions. We show that F_n,s(q,t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that F_n(q,t)=C_n(q,t) when t=1/q. We also show the partial symmetry relation F_n(q,1)=F_n(1,q). By modifying a proof of Haiman of a q-Lagrange inversion formula based on results of Garsia and Gessel, we obtain a q-analogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word.     

Some limit properties for the coefficients of the q-Catalan numbers

In the paper, some limit properties, such as, moderate deviation principle and law of large numbers, for the coefficients of the q-Catalan numbers and the generalized q-Catalan numbers are obtained.     

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q‐differential operator having the q‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.     

q-Catalan numbers

q-analogs of the Catalan numbers C_n = (1/(n + 1))(_n²ⁿ) are studied from the view-point of Lagrange inversion. The first, due to Carlitz, corresponds to the Andrews-Gessel-Garsia q-Lagrange inversion theory, satisfies a nice recurrence relation and counts inversions of Catalan words. The second, tracing back to Mac Mahon, arise from Krattenthaler's and Gessel and Stanton's q-Lagrange inversion formula, have a nice explicit formula and enumerate the major index. Finally a joint generalization is given which includes also the Polya-Gessel q-Catalan numbers.     

A semi-recursion for the number of involutions in special orthogonal groups over finite fields

Let I(n) be the number of involutions in a special orthogonal group SO(n,Fq) defined over a finite field with q elements, where q is the power of an odd prime. Then the numbers I(n) form a semi-recursion, in that for m>1 we haveI(2m+3)=(q2m+2+1)I(2m+1)+q2m(q2m−1)I(2m−2). We give a purely combinatorial proof of this result, and we apply it to give a universal bound for the character degree sum for finite classical groups defined over Fq.     

Representations of Central Matrix-Valued Carathéodory Functions in Both Nondegenerate and Degenerate Cases

We derive left and right quotient representations for central q × q matrix-valued Carathéodory functions. Moreover, we obtain recurrent formulas for the matrix polynomials involved in the quotient representations. These formulas are the starting point for getting recurrent formulas for those matrix polynomials which occur in the Arov-Krein resolvent matrix for the nondegenerate matricial Carathéodory problem.     

Pseudo q-Engel expansions and Rogers-Ramanujan type identities

Abstract

Andrews, Knopfmacher and Knopfmacher have used the Schur polynomials to consider the celebrated Rogers-Ramanujan identities in the context of q-Engel expansions. We extend this view using similar polynomials, provided by Sills, in the context of Slater's list of 130 Rogers-Ramanujan type identities.     

Weyl sums over integers with affine digit restrictions

For any given integer q?2, we consider sets N of non-negative integers that are defined by affine relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1's equals twice the number of 0's in the binary representation). The main goal is to prove that the sequence (αn)_n∈N is uniformly distributed modulo 1 for all irrational numbers α. The proof is based on a saddle point analysis of certain generating functions that allows us to bound the corresponding Weyl sums.     

Permutation polynomials over finite fields from a powerful lemma

Using a lemma proved by Akbary, Ghioca, and Wang, we derive several theorems on permutation polynomials over finite fields. These theorems give not only a unified treatment of some earlier constructions of permutation polynomials, but also new specific permutation polynomials over Fq. A number of earlier theorems and constructions of permutation polynomials are generalized. The results presented in this paper demonstrate the power of this lemma when it is employed together with other techniques.     

On the number of lines in planar spaces

SupposeS is a planar space withv>4 points and letq be the positive real number such thatv=q ³+q²+q+1. Assuming a weak non-degeneracy condition, we shall show thatS has at least (q²+1)(q²+q+1) lines with equality iffq is a prime power andS=PG(3,q).     

The q-Catalan Numbers: A Saddle Point Approach

Using the saddle point method, we obtain from the generating function of the q-Catalan numbers and Cauchy’s integral formula asymptotic results in central and non-central regions.     

Bivariate hermite interpolation and applications to Algebraic Geometry

Summary We try to solve the bivariate interpolation problem (1.3) for polynomials (1.1), whereS is a lower set of lattice points, and for theq-th interpolation knot,A _q is the set of orders of derivatives that appear in (1.3). The number of coefficients |S| is equal to the number of equations |A _q |. If this is possible for all knots in general position, the problem is almost always solvable (=a.a.s.). We seek to determine whether (1.3) is a.a.s. An algorithm is given which often gives a positive answer to this. It can be applied to the solution of a problem of Hirschowitz in Algebraic Geometry. We prove that for Hermite conditions (1.3) (when allA _q are lower triangles of orderp) andP is of total degreen, (1.3) is a.a.s. for allp=1, 2, 3 and alln, except for the two casesp=1,n=2 andp=1,n=4.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThis work has been partly supported by the Texas ARP and the Deutsche Forschungsgemeinschaft     

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.     

A filtration of -Catalan numbers

The operator of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999) 363–420] acting on the k-Schur functions [L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003) 103–146; L. Lapointe, J. Morse, Schur functions analogs for a filtration of the symmetric functions space, J. Combin. Theory Ser. A 101 (2003) 191–224; L. Lapointe, J. Morse, Tableaux on k+1-cores, reduced words for affine permutations and k-Schur expansion, J. Combin. Theory Ser. A 112 (2005) 44–81] indexed by a single column has a coefficient in the expansion which is an analogue of the (q,t)-Catalan number with a level k. When k divides n we conjecture a representation theoretical model in this case such that the graded dimensions of the module are the coefficients of the (q,t)-Catalan polynomials of level k. When the parameter t is set to 1, the Catalan numbers of level k are shown to count the number of Dyck paths that lie below a certain Dyck path with q counting the area of the path.     

Blocking Sets Of External Lines To A Conic In PG(2,q), q ODD

We determine all point-sets of minimum size in PG(2,q), q odd that meet every external line to a conic in PG(2,q). The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of PGL(2,q). * Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni and by the Hungarian-Italian Intergovernemental project “Algebraic and Geometric Structures”.     

The approximation by q-Bernstein polynomials in the case q ↓ 1

Let B_n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B_n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B_n (f, q_n; x)} with q_n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q_n ↓ 1. At the same time, there exists a sequence 0 < δ_n ↓ 0 such that the condition implies the approximation of f by {B_n (f, q_n; x)} for all f ∈C[0, 1]. Received: 15 March 2005