Generalization of Fibonomial Coefficients

Abstract

Following Lucas and then other Fibonacci people Kwaśniewski had introduced and had started ten years ago the open investigation of the overall F -nomial coefficients which encompass among others Binomial, Gaussian and Fibonomial coefficients with a new unified combinatorial interpretation expressed in terms of cobweb posets' partitions and tilings of discrete hyperboxes. In this paper, we deal with special subfamily of T -nomial coefficients.The main aim of this note is to develop the theory of T -nomial coefficients with the help of generating functions. The binomial-like theorem for T -nomials is delivered here and some consequences of it are drawn. A new combinatorial interpretation of T -nomial coefficients is provided and compared with the Konvalina way of objects' selections from weighted boxes. A brief summary of already known properties of F -nomial coefficients is served.

A note on Lucas L binomials and Ward Horadam binomials recurrences

“…Compare now: this paper recurrence formula (42) with recurrence formula (4) in [43,2010], compare this paper recurrence formula (43) with Proposition 2.2. in [43,2010] by Bruce E. Sagan and Carla D. Savage. Compare this paper recurrence (28) equivalent to (5) and proposition 2.2. in [43,2010] and note that ( 5) in [43,2010] is just the same -as ( 58) in [1,1878] -the same except for notation.…”

“…There might be various h 1 (r, s)(x) = h 1 and h 2 (r, s)(x) s = h 2 solutions of ( 30) and (21). Compare (38) in Example 1 with (42) in Example 3 below. As the possible h 1 (r, s)(x) = h 1 and h 2 (r, s)(x) s = h 2 formal solutions of ( 30) and ( 21) we may take…”

“…Compare (42) with equivalent recurrence (48) in order to notice that both h 1 and h 2 functions are different from case to case of recurrence (30) equivalent realizations.…”

A note on V-binomials recurrence for Lucas companion to $U_n$ sequence $V_n$

“…As for combinatorial interpretations of L-binomial or L-multinomial coefficients we leave that subject apart from this note because this note is to be deliberately short. Nevertheless we direct the reader to some papers and references therein; these are here the following: [32] (2010), [2] (2009), [31] (2009), [33] (2009 , [11] (1989), [13] (1991), [15] (1992), [16] (1993), [17] (1994), [18] (1994), [28] (2004), and to this end see [9]. The list is far not complete.…”