On multi F-nomial coefficients and Inversion formula for F-nomial coefficients

Abstract

In response to [7], we discover the looked for inversion formula for F -nomial coefficients. Before supplying its proof, we generalize F -nomial coefficients to multi F -nomial coefficients and we give their combinatorial interpretation in cobweb posets language, as the number of maximal-disjoint blocks of the form σP k 1 ,k 2 ,...,ks of layer Φ 1 → Φ n . Then we present inversion formula for F -nomial coefficients using multi F-nomial coefficients for all cobwebadmissible sequences. To this end we infer also some identities as conclusions of that inversion formula for the case of binomial, Gaussian and Fibonomial coefficients.

Generalization of Fibonomial Coefficients

“…For further reading about combinatorial interpretations we refer the reader to [6,8,13,14,15] and to the references therein.…”

Graded posets inverse zeta matrix formula

“…For the relevant recent developments see [7] while [8] is their all source paper as well as those reporting on the broader research (see [9][10][11][12][13][14][15][16][17][18][19][20][22][23][24][25][26] and references therein). The inspiration for "'philosophy"' of notation in mathematics as that in Knuth's from [21] -in the case of "'upside-downs"' has been driven by Gauss "'q-Natural numbers"'≡ N q = n q = q 0 + q 1 + ... + q n−1 n≥0 from finite geometries of linear subspaces lattices over Galois fields.…”

“…For that to deliver we use the Gaussian coefficients inherited upside down notation i.e. F n ≡ n F (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], [27][28][29][30],and the Appendix in [9] extracted from [32]) and recall the Upside Down Notation Principle.…”

“…F n ≡ n F . The Upside Down Notation Principle used since last century effectively (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and for earlier references therein; in particular see Appendix in [9] copied from [32]) may be formulated as a Principle i.e. trivial, powerful statement as follows.…”

On Cobweb Posets and Discrete F-Boxes Tilings

“…Here down in Figure 15 Cobweb Admissible sequences family A is defined in [9], GCD-morphic sequences family in [8]. Subfamily T λ of cobweb tiling sequences T is introduced in this note.…”