In this paper we study several problems concerning the number of homomor-phisms of trees. We begin with an algorithm for the number of homomorphisms from a tree to any graph. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching… (More)
We find a q-analog of the following symmetrical identity involving binomial coefficients n m and Eulerian numbers A n,m , due to Chung, Graham and Knuth [J. k≥0 a + b k A k,a−1 = k≥0 a + b k A k,b−1. We give two proofs, using generating function and bijections, respectively.
The papaya Y-linked region showed clear population structure, resulting in the detection of the ancestral male population that domesticated hermaphrodite papayas were selected from. The same populations were used to study nucleotide diversity and population structure in the X-linked region. Diversity is very low for all genes in the X-linked region in the… (More)
The (q, r)-Eulerian polynomials are the (maj−exc, fix, exc) enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical q-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic… (More)
We prove a conjecture of Gessel, which asserts that the joint distribution of descents and inverse descents on permutations has a fascinating refined γ-positivity.
We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind JS(n + k, n; z) by generalizing the analogous results for the Stir-ling and Legendre-Stirling numbers. More precisely, letting JS(n + k, n; z) = p k,0 (n) + p k,1 (n)z + · · · + p k,k (n)z k , we show that (1 − t) 3k−i+1 n≥0 p k,i (n)t n is a polynomial in t… (More)