753 research outputs found
Enumeration of rises and falls by position
AbstractLet π=(π1, π2,…,πn) denote a permutation of Zn = {1, 2,…, n}. The pair (πi, πi+1) is a rise if πi<πi+1 or a fall if πi>πi+1. Also a conventional rise is counted at the beginning of π and a conventional fall at the end. Let k be a fixed integer ≥ 1. The rise πi,πi+1 is said to be in a in a j (mod k) position if i ≡ j (mod k); similarly for a fall. The conventional rise at the beginning is in a 0 (mod k) position, while the conventional fall at the end is in an n (mod k) position. Let Pn≡Pn(r0,…,rk−1,ƒ0,…,ƒ;k−1) denote the number of permutations having ri rises i (mod k) positions and ƒ;i falls in i (mod k) positions. A generating function for Pn is obtained. In particular, for k = 2 the generating function is quite explicit and also, for certain special cases when k = 4
Normal Ordering for Deformed Boson Operators and Operator-valued Deformed Stirling Numbers
The normal ordering formulae for powers of the boson number operator
are extended to deformed bosons. It is found that for the `M-type'
deformed bosons, which satisfy , the
extension involves a set of deformed Stirling numbers which replace the
Stirling numbers occurring in the conventional case. On the other hand, the
deformed Stirling numbers which have to be introduced in the case of the
`P-type' deformed bosons, which satisfy , are found to depend on the operator . This distinction
between the two types of deformed bosons is in harmony with earlier
observations made in the context of a study of the extended
Campbell-Baker-Hausdorff formula.Comment: 14 pages, Latex fil
A double bounded key identity for Goellnitz's (big) partition theorem
Given integers i,j,k,L,M, we establish a new double bounded q-series identity
from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon
for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the
identity yields a strong refinement of Goellnitz's theorem with a bound on the
parts given by L. This is the first time a bounded version of Goellnitz's (big)
theorem has been proved. This leads to new bounded versions of Jacobi's triple
product identity for theta functions and other fundamental identities.Comment: 17 pages, to appear in Proceedings of Gainesville 1999 Conference on
Symbolic Computation
Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields
We present counting methods for some special classes of multivariate
polynomials over a finite field, namely the reducible ones, the s-powerful ones
(divisible by the s-th power of a nonconstant polynomial), and the relatively
irreducible ones (irreducible but reducible over an extension field). One
approach employs generating functions, another one uses a combinatorial method.
They yield exact formulas and approximations with relative errors that
essentially decrease exponentially in the input size.Comment: to appear in SIAM Journal on Discrete Mathematic
A note on q-Euler numbers and polynomials
The purpose of this paper is to construct q-Euler numbers and polynomials by
using p-adic q-integral equations on Zp. Finally, we will give some interesting
formulae related to these q-Euler numbers and polynomials.Comment: 6 page
On parity functions in conformal field theories
We examine general aspects of parity functions arising in rational conformal
field theories, as a result of Galois theoretic properties of modular
transformations. We focus more specifically on parity functions associated with
affine Lie algebras, for which we give two efficient formulas. We investigate
the consequences of these for the modular invariance problem.Comment: 18 pages, no figure, LaTeX2
The Wigner function associated to the Rogers-Szego polynomials
We show here that besides the well known Hermite polynomials, the q-deformed
harmonic oscillator algebra admits another function space associated to a
particular family of q-polynomials, namely the Rogers-Szego polynomials. Their
main properties are presented, the associated Wigner function is calculated and
its properties are discussed. It is shown that the angle probability density
obtained from the Wigner function is a well-behaved function defined in the
interval [-Pi,Pi), while the action probability only assumes integer values
greater or equal than zero. It is emphasized the fact that the width of the
angle probability density is governed by the free parameter q characterizing
the polynomial.Comment: 12 pages, 2 (mathemathica) figure
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