4 pages, presented at the Gian-Carlo Rota Polish Seminar, http://ii.uwb.edu.pl/akk/sem/sem_rota.htm, submitted to FECS'08: The 2008 International Conference on Frontiers in Education: WORLDCOMP'08
发表信息：
Proc. Jangjeon Math. Soc. volume 11 (1), 2008 (June),65-68
A research problem for undergraduates and graduates is being posed as a cap for the prior antecedent regular discrete mathematics exercises. [Here cap is not necessarily CAP=Competitive Access Provider, though nevertheless ...] The object of the cap problem of final interest i.e. array of fibonomial coefficients and the issue of its combinatorial meaning is to be found in A.K.Kwaśniewski's source papers. The cap problem number seven - still opened for students has been placed on Mathemagics page of the first author [http://ii.uwb.edu.pl/akk/dydaktyka/dyskr/dyskretna.htm]. The indicatory references are to point at a part of the vast domain of the foundations of computer science in ArXiv affiliation noted as CO.cs.DM. The presentation has been verified in a tutor system of communication with a couple of intelligent students. The result is top secret.Temporarily. [Contact: Wikipedia; Theory of cognitive development].
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,…
Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.
The explicit formula for mobiusien function of fibonacci cobweb poset P is given for the first time by the use of definition of P in plane grid coordinate system.
A complete characterization of two functions $f(x,y)$ and $g(x,y)$ in the $(f,g)$-inversion is presented. As an application to the theory of hypergeometric series, a general bibasic summation formula determined by $f(x,y)$ and $g(x,y)$ as well as four arbitrary sequences is obtained which unifies Gasper and Rahman's, Chu's and Macdonald's bibasic summation formula. Furthermore, an alternative proof of the $(f,g)$-inversion derived from the $(f,g)$-summation formula is presented. A bilateral $(f,g)$-inversion containing Schlosser's bilateral matrix inversion as a special case is also obtained.
We prove a recent conjecture of Blanco and Petersen (arXiv:1206.0803v2) about an expansion formula for inversions and excedances in the symmetric group.
The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in pa…
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 use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base $$\left\{z^n\frac{(az:q)_n}{(bz:q)_n}\bigg|n=0,1,2\cdots\right\}.$$ Some concrete expansion formulas and their applications to $q$-series identities are presented, including Carlitz's $q$-expansion formula, a new partial theta function identity and a coefficient identity of Ramanujan's ${}_1ψ_1$ summation formula as special cases.
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…
We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall-Littlewood symmetric functions.
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.
We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\to R_{\ge0}$ and associated generating function $P(t)=\sum_{m\in M}t^{deg(m)}$, called the growth function of $M$. We also introduce, using some towers of minimal common multiple sets in $M$, another signed generating function $N(t)$, called the skew-growth function of $M$. We show that these functions satisfy the inversion formula $P(t)N(t)=1$. In case the monoid is the set of positive integers with ordinary product structure and the degree map is logarithm function, using the coordinate change $t=exp(-s)$, the…
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.
We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the left adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilised to provide examples showing limitations of two standard tools of the Koszul duality theory. We also apply these functors to get a homotopical algebra proof of the Lagrange inversion formula.
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic which is composed of three parts and which is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a geometric description of the statistic in the style of the classical case. We go on to show that in the n x (n+1) lattice, parki…
Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we present a combinatorial proof of the inversion formula on $R$-polynomials due to Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer's combinatorial interpretation of the $R$-polynomials in terms of increasing Bruhat paths, we reformulate the inversion formula in terms of $V$-paths. By a $V$-path from $u$ to $v$ with bottom $w$ we mean a pair $(Δ_1,Δ_2)$ of Bruhat paths such that $Δ_1$ is a decreasing path from $u$ to $w$ and $Δ_2$ i…
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…
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 …
This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants of even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again this formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single pr…
In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain augmented double Segal spaces subject to some exactness conditions. We establish a Möbius inversion principle for (co)modules, and a Rota formula for certain more involved structures called Möbius bicomodule configurations. The most important instance of the latter notion arises as mapping cylinders of infinity adjunctions, or more generally of …
We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.
We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials…
Inversion formulas have been found, converting between Stirling, tanh and Lah numbers. Tanh and Lah polynomials, analogous to the Stirling polynomials, have been defined and their basic properties established. New identities for Stirling and tangent numbers and polynomials have been derived from the general inverse relations. In the second part of the paper, it has been shown that if shifted-gamma probability densities and negative binomial distributions are matched by equating their first three semi-invariants (cumulants), then the cumulants of the two distributions are related by a pair of r…
Bjoerner and Wachs provided two q-generalizations of Knuth's hook formula counting linear extensions of forests: one involving the major index statistic, and one involving the inversion number statistic. We prove a multivariate generalization of their inversion number result, motivated by specializations related to the modular invariant theory of finite general linear groups.
We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function. The unique explicit expression for the coefficients that appears to be known is that in the book by L. Comtet, which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet's expr…
We extend the author's formula (2011) of weighted counting of inversions on permutations to the one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix recently shown by Brualdi-Schroeder and the author (both 2017) independently.
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.
As further development of earlier works on the $(f,g)$-inversion, the present paper is devoted to the $(f,g)$-difference operator and the representation problem or an expansion formula of analytic functions. A recursive formula and the Leibniz formula for the $(f,g)$-difference operator of the product of two functions are established. The resulting expansion formula not only unifies the $q$-analogue of the Lagrange inversion formula of Gessel and Stanton (thus, a $q$-expansion formula of Liu) for $q$-series but also systematizes the "Ismail's argument". In the meantime, a rigorous analytic pro…
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.
