In this paper, we develop determinantal representations for generalised Laguerre polynomials, expressing them in terms of Toeplitz–Hessenberg matrices, first for non-negative integer parameters and then for arbitrary real parameters. Using Bell polynomials, we extend this framework to Sheffer and Jacobi polynomials and obtain an alternative Hessenberg representation for generalised Laguerre polynomials, together with analogous formulas for several related polynomial families. Finally, continued-fraction expansions are used to derive combinatorial identities associated with these determinants, revealing connections between orthogonal polynomials and combinatorial structures.
(Im)balance indices can be used to quantify the (im)balance of trees by assigning numerical scores to them. An easy way to generate a new index is to construct a compound index, e.g., a linear combination of established indices. Two of the most prominent and widely used imbalance indices are the Sackin index and the Colless index. In this study, we show that these classic indices are themselves compound in nature: they can be decomposed into more elementary components that independently satisfy the defining properties of a tree (im)balance index. We further show that the difference Colless minus Sackin results in another imbalance index that is minimized (among others) by all Colless minimal trees. Conversely, the difference Sackin minus Colless forms a balance index. Finally, we compare the building blocks of which the Sackin and the Colless indices consist to these indices as well as to the stairs2 index, which is another index from the literature. Our results suggest that the elementary building blocks we identify are not only foundational to established indices but also valuable tools for analyzing disagreement among indices when comparing the balance of different trees. Along the way, we investigate the so-called echelon tree, which plays an important role for several (im)balance indices, and present the first non-recursive algorithm to construct it.
Let K be a finite r-dimensional simplicial complex on n vertices. Motivated by classical bounds for the largest Laplacian eigenvalue of a graph, we study the largest eigenvalue of the combinatorial (r-1) -up Laplacian L^up_r-1(K) . We first give a topological equality criterion for the universal bound: λ _max (L^up_r-1(K) )≤ n. Equality holds if and only if the r-dimensional complement K^c has nonzero reduced homology H_r-1(K^c,ℝ) . For r=1 , this is precisely the classical condition that the complement graph is disconnected. We then prove the local bound λ _max (L^up_r-1(K) ) ≤max _F∈ S_r(K) |⋃ _E∈∂ FN_K(E) | ≤ n, where N_K(E) is the set of vertices that complete the (r-1) -face E to an r-face of K. This extends Das’ neighborhood-union bound for graphs and sharpens the known degree-sum bound for simplicial complexes. The proof decomposes the r-down Laplacian into local signed matrices indexed by (r+2) -vertex sets and yields an equality criterion through the kernel of a local coboundary-type map. We also construct multipartite semiregular complexes with admissible additions that attain the bound, and show that in higher dimensions these examples do not exhaust all equality cases.
We give a new proof of the formula of key polynomials in terms of semistandard key tableaux, originally obtained by Assaf [1, 2]. We show that the generating functions of semistandard key tableaux satisfy the recursive relations with respect to the action of Demazure operators. Our proof is inspired by the work of Wachs [11].
