Coding multitype forests: Application to the law of the total population of branching forests

Abstract

Abstract. By extending the breadth rst search algorithm to any d-type critical or subcritical irreducible branching forest, we show that such forests may be encoded through d independent, integer valued, d-dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem which is proved here, allow us to give an explicit form of the law of the total progeny, jointly with the number of subtrees of each type, in terms of the ospring distribution of the branching process. We then apply these results to some enumeration formulas of multitype forests with given degrees and to a new proof of the Lagrange-Good inversion Theorem.

Critical Multi-type Galton–Watson Trees Conditioned to be Large

“…The proof of Theorem 3.1 relies on two arguments. The first one is a generalization of the Dwass formula for multi-type GW processes given by Chaumont and Liu [5] which encodes critical or sub-critical d-multi-type GW forests using d random walks of dimension d. The second one is the strong ratio theorem for random walks in Z d , see Theorem 4.7, which generalizes a result by Neveu [18] in dimension one. The proof of the strong ratio theorem relies on a uniform version of the d-dimensional local theorem of Gnedenko [7], see also Gnedenko and Kolmogorov [8] (for the sum of independent random variables), Rvaceva [22] (for the sum of d-dimensional i.i.d.…”

Characterizing the Transmission Potential of Zoonotic Infections from Minor Outbreaks

“…Pre-existing immunity in older age groups can alter this pattern [22], making it possible to separate the reproduction number into its pathogen-and population-specific components. We made use of this observation by developing a novel agestructured model of stuttering transmission chains, which combined reported social contact data with a multi-type branching process [23,24].…”

“…If m > 0, we had[24], Pðn; m; a 21 ; a 12 Þ ¼ a 21 T n 11 ðn À a 12 À 1ÞT n 21 ða 21 ÞT m 12 ða 12 ÞT m 22 ðm À a 21 Þ nm :…”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…We show that certain specializations of that reduced polynomial coincide, among others, with the Grothendieck polynomials corresponding to the permutation 1 × w (n−1) 0 ∈ S n , the Lagrange inversion formula, as well as give rise to combinatorial (i.e., positive expressions) multiparameters deformations of Catalan and Fuss-Catalan, Motzkin, Riordan and Fine numbers, Schröder numbers and Schröder trees. We expect (work in progress) a similar connections between Schubert and Grothendieck polynomials associated with the Richardson permutations 1 k × w (n−k) 0 , k-dissections of a convex (n + k + 1)-gon investigated in the present paper, and k-dimensional Lagrange-Good inversion formula studied from combinatorial point of view, e.g., in [22,50].…”

“…22. The value N n (4) of the Narayana polynomial at β = 4 has the following combinatorial interpretation: N n (4) is equal to the number of different lattice paths from the point (0, 0) to that (n, 0) using steps from the set Σ = {(k, k) or (k, −k), k ∈ Z >0 }, that never go below the x-axis, see [131, A059231].…”