The so-called ζ-analogues of the Stirling numbers of the first and second kind are considered. These numbers cover ordinary binomial and Gaussian coefficients, p, qStirling numbers and other combinatorial numbers studied with the help of object selection, Ferrers diagrams and rook theory. Our generalization includes these and now also the p, q-binomial coefficients. This special subfamily of F ...

Abstract The excedance set of a permutation π = π1π2 · · ·πk is the set of indices i for which πi > i. We give explicit formulas for the number of permutations whose excedance set is the initial segment {1, 2, . . . ,m} and also of the form {1, 2, . . . ,m,m + 2}. We provide two proofs. The first is an explicit combinatorial argument using rook placements. The second uses the chromatic polynomi...

The spatial weighting matrix is very important to overview of the relationship between one location another in regression. In this study, authors compare queen contiguity and rook SAR SEM models stunting cases Bone Bolango Regency, Gorontalo Province. variables used are number IDL, percentage LBW, amount proper sanitation, exclusively breastfed babies, poor people. purpose study was determine f...

In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S(α, β, r) and S(α, β, r) as introduced by Hsu and Shiue [17]. We show that in the case where β = 0 and α and r are nonnegative integers both of our (p, q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We al...

In the 1920’s Artin defined the braid group, Bn, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most ...

Androgenetic alopecia in the female occurs much more frequently than is generally believed. The condition is still considered infrequent, for it differs, in its clinical picture and in the sequence of events leading to it, from common baldness in men. To facilitate an early diagnosis (desirable in view of the therapeutic possibilities by means of antiandrogens) a classification of the stages of...

We provide a structural description of Bruhat order on the set F2n of fixed-pointfree involutions in the symmetric group S2n which yields a combinatorial proof of a combinatorial identity that is an expansion of its rank-generating function. The decomposition is accomplished via a natural poset congruence, which yields a new interpretation and proof of a combinatorial identity that counts the n...