In a Galton-Watson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...

Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...

Abstract. A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials n−1 Pn(a, b, c) = c (ia + (n − i)b + c), i=1 which reduce to to (n + 1) n−1 for a = b = c = 1. Our study of proper vertices was motivated by Postnikov’s hook length formula (n + 1) n−1 = n!

Abstract — We introduce join scheduling algorithms that employ a balanced network utilization metric to optimize the use of all network paths in a global-scale database federation. This metric allows algorithms to exploit excess capacity in the network, while avoiding narrow, long-haul paths. We give a twoapproximate, polynomial-time algorithm for serial (left-deep) join schedules. We also present extensions to this algorithm that explore parallel schedules, reduce resource usage, and define tradeoffs between computation and network utilization. We evaluate these techniques within the SkyQuery federation of Astronomy databases using spatial-join queries submitted by SkyQuery’s users. Experiments show that our algorithms realize near-optimal network utilization with minor computational overhead. I.

. In this paper we are interesting in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let Tn;k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of the fact that jTn+1;k j = \Gamma n k \Delta n n\Gammak . Moreover, we use the family Tn+1;0 of Cayley trees for which the root is smaller than all its sons to give combinatorial explanations of various identities involving n n . We rely this family to the enumeration of minimal factorization of the n-cycle (1; 2; : : : ; n) as a product of transpositions. Finally, we use the fact that jTn+1;0 j = n n to prove bijectively that there are 2n n ordered alternating trees on [n + 1]. R' esum' e. Dans cet article nous nous int'eressons `a l"enum'eration d'arbres 'etiquet'es enracin'es, en consid'erant un nouveau param`etre relatif `a l'ordre existant entre la racine et ses fils. Soit donc Tn;k la famille d...

This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating out-degrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1

Abstract. The Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. These polynomials have been closely related to the enumeration of trees. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the gramatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor’s. Furthermore, Shor introduces a sequence of polynomials generalizing the numbers mentioned above. It was a great coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polynomials through an explicit substitution of parameters. Moreover, Zeng gives two combinatorial interpretations of the recurrence relation of Shor. On the other side of the story, Shor also discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt-Evans-Wilson recursion under the substitution of Zeng, and asks for a combinatorial interpretation. The objective of this paper is to present a bijection for the Shor recursion, or and Berndt-Evans-Wilson recursion, answering the question of Shor. Such a bijection also leads to a combinatorial interpretation of the recurrence relation originally given by Ramanujan.

Abstract. The Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the grammatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor’s. It was a coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polynomials through an explicit substitution of parameters. Shor also discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt-Evans-Wilson recursion under the substitution of Zeng, and asks for a combinatorial interpretation. The objective of this paper is to present a bijection for the Shor recursion, or and Berndt-Evans-Wilson recursion, answering the question of Shor. Such a bijection also leads to a combinatorial interpretation of the recurrence relation originally given by Ramanujan. 1

Abstract.Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Qn: = Qn(x,y,z,t) defined by Q1 = 1, Qn+1 = [x + nz + (y + t)(n + y∂y)]Qn. In this paper we prove Chapoton’s conjecture on the duality formula: Qn(x,y,z,t) = Qn(x+nz+ nt,y, −t, −z), and answer his question about the combinatorial interpretation of Qn. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees.