First Remark on a ζ-Analogue of the Stirling Numbers

Abstract

The so-calledOur generalization includes these and now also the

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

“…8. Maciej Dziemiańczuk in [155,2011] extends the results of John Konvalina from 4. above. The Dziemiańczuk' ζ -analogues of the Stirling numbers arrays of both kinds cover ordinary binomial and Gaussian coefficients, p, q-Stirling numbers and other combinatorial numbers studied with the help of object selection, Ferrers diagrams and rook theory.…”

A note on Lucas L binomials and Ward Horadam binomials recurrences

“…8. Maciej Dziemiańczuk in [146,2011] extends the results of John Konvalina from 4. above. The Dziemiańczuk' ζ -analogues of the Stirling numbers arrays of both kinds cover ordinary binomial and Gaussian coefficients, p, q-Stirling numbers and other combinatorial numbers studied with the help of object selection, Ferrers diagrams and rook theory.…”

Dually weighted Stirling-type sequences

“…Another specific case is the ζ-analogue of Stirling numbers introduced in [5]. These are also V-Stirling numbers that are obtained by letting V = (ζ i , w i−1 ).…”

“…As an illustration, let V = (2i+ 4, 1), and φ 0,1 V and ψ 0,1 V be the 0, 1 v -tableaux in Figure 1, where φ 0,1 V is of shape 3 1 1 2 4 4 ∈ T 0,0 [5,3] and ψ 0,1 V is of shape 3 1 0 2 4 5 ∈ T d 0,0 [5,3]. The lattice path associated with φ 0,1 V is V V HHV V HV V V .…”

“…However, we can also apply the same method of obtaining a permutation or partition to the boxes below the lattice path. Recall that the b-Stirling number of the first kind are defined by the generating function in (5) and are obtained by letting V = (i, i) and α = β = 0. They can be combinatorially interpreted as the number of pairs of permutations (σ 1 , σ 2 ), where σ 1 is a permutation of [n] into disjoint cycles C 1 , C 2 , .…”