Reduction trees are a way of encoding a substitution procedure dictated by the relations of an algebra. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We explain how a shelling of the canonical triangulation can be read off from the corresponding reduction tree in the subdivision algebra. We then introduce the notion of shellable reduction trees in the subdivision and related algebras and define h-polynomials of reduction trees. In the case of the subdivision algebra, the h-polynomials of the canonical triangulations of flow polytopes equal the h-polynomials of the corresponding reduction trees, which motivated our definition. We show that the reduced forms in various algebras, which can be read off from the leaves of the reduction trees, specialize to the shifted h-polynomials of the corresponding reduction trees. This yields a technique for proving nonnegativity properties of reduced forms. As a corollary we settle a conjecture of A.N. Kirillov.
We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other…
By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma-Mansour-Wang-Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin-Ma-Ma-Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of "even-degree nodes on odd levels" and "odd-degree nodes" on weakly increasing trees is found, extending the Schett polynomials, a generalizatio…
Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.
We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for the joint distribution of descents, big ascents and successions of permutations. And then, inspired by the work of Chen and Fu on the trivariate second-order Eulerian polynomials, we show the e-positivity of the multivariate k-th order Eulerian polynomials, which are the enumerators for the joint distribution of ascents, descents and j-plateaux of k-Stirling…
We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing sequence determines an algorithm which can be applied to a single function, or to the set $\mathcal{P}_{G,q}$ of $G$-parking functions. When the latter is chosen, the algorithm uses splitting operations - inspired by the recursive defintion of the Tutte polynomial - to iteratively break $\mathcal{P}_{G,q}$ into disjoint subsets. This results in bijective map…
A graph $G$ on $n$ vertices of diameter $D$ is called $H$-palindromic if $α(G,k) = α(G,D-k)$ for all $k=0, 1, \dots, \left \lfloor{\frac{D}{2}}\right \rfloor$, where $α(G,k)$ is the number of unordered pairs of vertices at distance $k$. Quantities $α(G,k)$ form coefficients of the Hosoya polynomial. In 1999, Caporossi, Dobrynin, Gutman and Hansen showed that there are exactly five $H$-palindromic trees of even diameter and conjectured that there are no such trees of odd diameter. We prove this conjecture for bipartite graphs. An infinite family of $H$-palindromic trees of diameter $6$ is also …
We present a polynomial time reduction from the multi-graph isomorphism problem to the problem of code equivalence of additive codes over finite extensions of ${\mathbb F}_2$.
On Unimodality of Independence Polynomials of some Well-Covered Trees
图$G$中基数$k$的稳定集的个数是图$G$的独立多项式的第$k$个系数（I. Gutman和F. Harary，1983）。1990年，Y. O. Hamidoune证明了对于任何无爪图，它的独立多项式是单峰的，即，存在系数$k$，使得系数序列中从第一个到第$k$个的部分不减，而系数的第二部分不增。Y. Alavi，P. J. Malde，A. J. Schwenk和P. Erdös（1987）提出了一个问题：树（或者森林）的独立多项式是否是单峰的。J. I.布朗，K。Dilcher和R. J. Nowakowski（2000）证明了这对任何覆盖良好的图（所有最大独立集具有相同大小的图）都是正确的。V.E. Levit和E. Mandrescu（1999）证明了每一个良好覆盖的树都可以作为许多良好覆盖的蜘蛛的连接来获得，其中蜘蛛是一个最多有一个度至少为3的顶点的树。在本文中，我们证明了任何覆盖良好的蜘蛛的独立多项式是单峰的。此外，我们还介绍了一些关于独立多项式的图变换。它们使我们能够减少几种类型的覆盖良好的树无爪图，因此，证明他们的独立多项式是单峰的。
We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and trees on $n+1$ vertices where $k$ children of the root are smaller than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials …
