Haibao Duan
We present both a combinatorial characterization and a recurrent formula for the entries of the inverse Kostka matrix. An application to the topology of the classifying space BU(n) is obtained.
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A combinatorial interpretation of the noncommutative inverse Kostka matrix
该研究提出无瑕非交换对称函数展开为完全齐次非交换对称函数的组合公式，引入GBPR图与隧道钩，推广至斜形状及整数序列索引的无瑕函数，并拓展了Campbell的带状分解结果。
A new proof of an Eğecioğlu--Remmel inverse Kostka matrix problem via a Garsia--Milne involution involving Sym and NSym
本文通过引入隧道钩覆盖和反向双射，给出了Eğecioğlu--Remmel逆Kostka矩阵问题的全新证明，不同于Loehr和Mendes的结果。
We study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by Knutson and Tao in their work on the equivariant cohomology of the Grassmannian.
$e$-positivity of chromatic quasisymmetric functions of bounce number three via the inverse Kostka matrix
本文利用逆Kostka矩阵的组合解释，在反弹数为三的Dyck路径下展开色拟对称函数，证明其$e$-正性，并推广至含$n-3$元素反链的序。
Let $K(q,t)= \|K_{\laμ}(q,t)\|_{\la,μ}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\laμ}(t). In this paper we present a new proof of the polynomiality of the q,t-Kostka coefficients that is both short and elementary. More precisely, we derive that $K(q,t)$ has entries in \ZZ[q,t] directly from the fact that the matrix $K(t)^{-1}$ has entries in \ZZ[t]. The proof uses only identities that can be found in the original paper [7] of Macdonald.
A local framework for proving combinatorial matrix inversion theorems
本文提出递归组合结构下矩阵互逆的局部化证明框架，并以 Kostka 矩阵、对称群特征标表等应用给出新的规范双射证明。
We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal inversion" statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q=1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the constru…
The vertex-edge incidence matrix of a (connected) unicyclic graph G is a square matrix which is invertible if and only if the cycle of G is an odd cycle. A combinatorial formula of the inverse of the incidence matrix of an odd unicyclic graph was known. A combinatorial formula of the Moore-Penrose inverse of the incidence matrix of an even unicyclic graph is presented solving an open problem.
We show that certain weighted average of the alpha-determinant of a $kn$ by $kn$ matrix of the form $A\otimes1_{1,k}$, the Kronecker product of a $kn$ by $n$ matrix $A$ and $1$ by $k$ all one matrix $1_{1,k}$, over permutations of $kn$ letters is reduced to the $k$-wreath determinant of $A$ up to constant. The constant is exactly given by the modified content polynomial for the Young diagram $(k^n)$. As a corollary, we give a `determinantal' formula for certain functions on the symmetric groups which are invariant under the left and right translation by a Young subgroup, especially the values …
Inverse Formulae for $q$-analogues of Bipartite Distance Matrix
本文研究二部距离矩阵的两种 q-类似物，给出其逆矩阵公式，并由此引入二部拉普拉斯矩阵的 q-类似版本。
Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Egecioglu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and …
A connected graph is called a multi-block graph if each of its blocks is a complete multi-partite graph. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the distance matrix for a class of multi-block graphs.
A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative matrix realization of $G$ is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. Similar to trees, we obtain a relation for the inverse of the distance matrix of a class of completely positive graphs involving the Laplacian matrix, a r…
In this paper we provide the group inverse of the combinatorial Laplacian matrix of distance-biregular graphs using the so-called equilibrium measures for sets obtained by deleting a vertex. We also show that the two equilibrium arrays characterizing a distance-biregular graph can be expressed in terms of the mentioned equilibrium measures. As a consequence of the minimum principle, we show a characterization of when the group inverse of the combinatorial Laplacian matrix of a distance-biregular graph is an $M$-matrix.
On the Frobenius norm of the inverse of a non-negative matrix
本文证明非负矩阵逆的 Frobenius 范数的新下界，并据此在维度超过小常数时完全解决 S-矩阵猜想。
In this paper, we give necessary and sufficient conditions for a real symmetric matrix, and in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86--104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetri…
In this paper, we found the Moore-Penrose generalized inverse of adjacency matrix of an undirected graph, explicitly. We proved that the matrix $R_λ= [r_{ij}]$ is nonsingular where $r_{ii}=\frac{1}λ+ °v_i$ and $r_{ij}=\mid N_G(V_i)\cap N_G(V_j)\mid$ for $i\neq j$, and we proved that $A^{\dagger}_G=[s_{ij}]_{1\leq i, j \leq n}$ where $\displaystyle{s_{ij}=s_{ji}=\lim_{λ\rightarrow +\infty} \langle R^{-1}_λe_j, f_i \rangle }$. The proof of the main result was based on the Tikhonov regularization.
We suggest three applications for the inverses: For the inverse Motzkin matrix we look at Hankel determinants, and counting the paths inside a horizontal band, and for the inverse Schröder matrix we look at the paths inside the same band, but ending on the top side of the band.
We derive the explicit formula for the inverse of zeta matrix for any graded posets with the finite set of minimal elements . The combinatorial interpretation of this result is given. For that to do special number theoretic code triangles for graded posets are proposed and apart from the present author combinatorial interpretation of $F-nomial$ coefficients another one is proposed referring to the number of all maximal chains in the corresponding poset intervals.
Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in …
If a graph has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a tree is non-singular if and only if the tree has a unique perfect matching; in this case the determinant of the matrix turns out to be $\pm 1$ and the inverse of the tree was shown to be `switching-equivalent' to a simple graph [C. Godsil, Inverses of Trees, Combinatorica 5 (1985), 33--39]. Using generalized inverses of symmetric matrices (that coincide with Moore-Penro…
During routine state space circuit analysis of an arbitrarily connected set of nodes representing a lossless LC network, a matrix was formed that was observed to implicitly capture connectivity of the nodes in a graph similar to the conventional incidence matrix, but in a slightly different manner. This matrix has only 0, 1 or -1 as its elements. A sense of direction (of the graph formed by the nodes) is inherently encoded in the matrix because of the presence of -1. It differs from the incidence matrix because of leaving out the datum node from the matrix. Calling this matrix as forward adjac…
We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $BC_n$ Koornwinder polynomials $P_{(1^r)}(x|a,b,c,d|q,t)$ with one column diagrams, to the type $BC_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from $BC_n$ to $BC_{n-1}$. To have a good description of the transition matrices involved, we introduce the following degeneration schem…
Let $X$ be bipartite mixed graph and for a unit complex number $α$, $H_α$ be its $α$-hermitian adjacency matrix. If $X$ has a unique perfect matching, then $H_α$ has a hermitian inverse $H_α^{-1}$. In this paper we give a full description of the entries of $H_α^{-1}$ in terms of the paths between the vertices. Furthermore, for $α$ equals the primitive third root of unity $γ$ and for a unicyclic bipartite graph $X$ with unique perfect matching, we characterize when $H_γ^{-1}$ is $\pm 1$ diagonally similar to $γ$-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a n…
The reciprocal Pascal matrix is the Hadamard inverse of the symmetric Pascal matrix. We show that the ordinary matrix inverse of the reciprocal Pascal matrix has integer elements. The proof uses two factorizations of the matrix of super Catalan numbers.
In this paper, we consider matrices whose inverses are tridiagonal Z--matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal Z--matrix if and only if, up to a positive scaling of the rows, it is the Hadamard product of a so called weak type $\D$ matrix and a flipped weak type $\D$ matrix whose parameters satisfy certain quadratic conditions. We predict from these parameters to which class of Z--matrices the inverse belongs to. In particular, we give a characterization of inverse tridiagonal M--matri…
We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries' natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse.
A family of orthogonal functions on the unit circle and a new multilateral matrix inverse
本文利用Bailey求和证明了特定级数构成单位圆上的正交函数族，并从Dougall求和提取出新的双边矩阵逆，结合Pfaff-Saalschütz求和推导了 $_3H_3$ 级数求和公式及其多变量扩展。
We study the inverse problem of recovering a tree graph together with the weights on its edges (equivalently a metric tree) from the knowledge of the Dirichlet-to-Neumann matrix associated with the Laplacian. We prove an explicit formula which relates this matrix to the pairwise weighted distances of the leaves of the tree and, thus, allows to recover the weighted tree. This result can be viewed as a counterpart of the Calderón problem in the analysis of PDEs. In contrast to earlier results on inverse problems for metric graphs, we only assume knowledge of the Dirichlet-to-Neumann matrix for a…
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