In response to [6], 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 sP_{k_1,k_2,...,k_s} of layer <Phi_1-->Phi_n>. Then we present inversion formula for F-nomial coefficients using multi F-nomial coefficients for all cobweb-admissible sequences. To this end we infer also some identities as conclusions of that inversion formula for the case of binomial, Gaussian and Fibonomial coefficients.
In light of the well-known fact that the $n$th divided difference of any polynomial of degree $m$ must be zero while $m<n$,the present paper proves the $(α,β)$-inversion formula conjectured by Hsu and Ma [J. Math. Res. $\&$ Exposition 25(4) (2005) 624]. As applications of $(α,β)$-inversion, we not only recover some known matrix inversions due to Gasper, Schlosser, and Warnaar, but also fin three new matrix inversions related to elliptic divisibility sequence and theta functions.
In this paper, by means of the classical Lagrange inversion formula, we establish a general nonlinear inverse relations which is a partial solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8. (https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.pdf). As applications of this inverse relation, we not only find a short proof of another nonlinear inverse relation due to Birmajer et al., but also set up a few convolution identities concerning the Mina polyn…
The study of pattern avoidance in inversion sequences recently attracts extensive research interests. In particular, Zhicong Lin and Jun Ma conjectured a formula that counts the number of inversion sequences avoiding the pattern $0012$. We will not only confirm this conjecture but also give a formula that enumerates the number of $0012$-avoiding inversion sequences in which the last entry equals $n-1$.
Schubert polynomials $\mathfrak{S}_w$ are polynomial representatives for cohomology classes of Schubert varieties in a complete flag variety, while Grothendieck polynomials $\mathfrak{G}_w$ are analogous representatives for the $K$-theory classes of the structure sheaves of Schubert varieties. In the special case that $\mathfrak{S}_w$ is a multiplicity-free sum of monomials, K. Mészáros, L. Setiabrata, and A. St. Dizier conjectured that $\mathfrak{G}_w$ can be easily computed from $\mathfrak{S}_w$ via Möbius inversion on a certain poset. We prove this conjecture.
With the use of the $(f,g)$-matrix inversion under specializations that $f=1-xy,g=y-x$, we establish an $(1-xy,y-x)$-expansion formula. When specialized to basic hypergeometric series, this $(1-xy,y-x)$-expansion formula leads us to some expansion formulas expressing any ${}_{r}ϕ_{s}$ series in variable $x~t$ in terms of a linear combination of ${}_{r+2}ϕ_{s+1}$ series in $t$, as well as various specifications. All these results can be regarded as common generalizations of many konwn expansion formulas in the setting of $q$-series. As specific applications, some new transformation formulas of …
We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.
Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combi…
We introduce the Hopf monoid of sets of cycles and paths, which contains the Faà di Bruno Hopf monoid as a submonoid. We give cancellation-free and grouping-free formulas for its antipode, one in terms of tubings and one in terms of \emph{pointed} noncrossing partitions. We provide an explicit description of the group of characters of this Hopf monoid in terms of pairs of power series. Using graph associahedra, we relate paths and cycles to associahedra and cyclohedra, respectively. We give formulas for inversion in the group of characters in terms of the faces of these polytopes.
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.
