Related papers: A multivariate "inv" hook formula for forests
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some…
We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant…
We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We…
We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas…
We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur…
A result of Foata and Schutzenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property:…
Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han's formulas has been generalized to k-ary trees by Yang. Sagan has found a probabilistic proof of Yang's extension. We give…
We present a simple combinatorial proof of Postnikov's hook length formula for binary trees.
A hyperbinary expansion of a positive integer n is a partition of n into powers of 2 in which each part appears at most twice. In this paper, we consider a generalization of this concept and a certain statistic on the corresponding set of…
Two well known mahonian statistics on words are the inversion number and the major index. In 1996, Foata and Zeilberger introduced generalizations, parameterized by relations, of these statistics. In this paper, we study the statistics…
Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue of rook placements on a board. We develop this $q$-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding…
Bj\"orner and Wachs defined a major index for labeled plane forests and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests.…
This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out…
Lajos Takacs gave a somewhat formidable alternating sum formula for the number of forests of unrooted trees on $n$ labeled vertices. Here we use a weight-reversing involution on suitable tree configurations to give a combinatorial…
We study numerical invariants associated with the reduction of singularities of holomorphic foliation germs on $(\mathbb{C}^2, 0)$. Building on our previous work on generalized curve foliations, we extend explicit formulas for several…
Recently, a new weighted generalization of the branching rule for the hook lengths, equivalent to the hook formula, was proved. In this paper, we generalize the complementary branching rule, which can be used to prove Burnside's formula. We…
For words of length $n$, generated by independent geometric random variables, we consider the mean and variance of the number of inversions and of a parameter of Knuth from permutation in situ. In this way, $q$--analogues for these…
We develop direct bijections between the set $F_n^k$ of minimal factorizations of the long cycle $(0\,1\,\cdots\, kn)$ into $(k+1)$-cycle factors and the set $R_n^k$ of rooted labelled forests on vertices $\{1,\ldots,n\}$ with edges…
We consider, following the work of S. Kerov, random walks which are continuous-space generalizations of the Hook Walks defined by Greene-Nijenhuis-Wilf, performed under the graph of a continual Young diagram. The limiting point of these…
Block and G\"ottsche have defined a $q$-number refinement of counts of tropical curves in $\mathbb{R}^2$. Under the change of variables $q=e^{iu}$, we show that the result is a generating series of higher genus log Gromov-Witten invariants…