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We study lattice paths in the cube, starting at $(n,n,n)$ and ending at $(0,0,0)$, with unit steps $(-1,0,0)$, $(0,-1,0)$, $(0,0,-1)$. Our main interest is the number of times the diagonal $x=y=z$ is visited during the random walk. We derive the corresponding generating function of such lattice paths. We also turn to the cuboid, enumerating lattice paths starting at $(n_1,n_2,n_3)$ and ending at $(0,0,0)$ according to the visits to the cube's diagonal. Furthermore, we provide for the cube a refined enumeration according to visits after a detour of a certain length. These enumerations allow us to obtain distributional results for the corresponding random variables. Extensions to hypercube are discussed, as well as a summary of known results for the square. We collect applications to the sampling without replacement urn and a card guessing game. Finally, we also show how to recover (and extend) a very recent result of Li and Starr on Dyck bridges using generating functions and composition schemes., Comment: 20 pages and five figures

This work is devoted to P\'olya-Young urns, a class of periodic P\'olya urns of importance in the analysis of Young tableaux. We provide several extension of the previous results of Banderier, Marchal and Wallner [Ann. Prob. (2020)] on P\'olya-Young urns and also generalize the previously studied model. We determine the limit law of the generalized model, involving the the local time of noise-reinforced Bessel processes. We also uncover a martingale structure, which leads directly to almost-sure convergence of the random variable of interest. This allows us to add second order asymptotics by providing a central limit theorem for the martingale tail sum, as well as a law of the iterated logarithm. We also turn to random vectors and obtain the limit law of P\'olya-Young urns with multiple colors. Additionally, we introduce several growth processes and combinatorial objects, which are closely related to urn models. We define increasing trees with periodic immigration and we related the dynamics of the P\'olya-Young urns to label-based parameters in such tree families. Furthermore, we discuss a generalization of Stirling permutations and obtain a bijection to increasing trees with periodic immigration. Finally, we introduce a chinese restaurant process with competition and relate it to increasing trees, as well as P\'olya-Young urns., Comment: 35 pages, 7 figures, 1 table

Early investigation of P\'{o}lya urns considered drawing balls one at a time. In the last two decades, several authors considered multiple drawing in each step, but mostly for schemes on two colors. In this manuscript, we consider multiple drawing from urns of balls of multiple colors, formulating asymptotic theory for specific urn classes and addressing more applications. The class we consider is affine and tenable, built around a "core" square matrix. An index for the drawing schema is derived from the eigenvalues of the core. We identify three regimes: small-, critical-, and large-index. In the small-index regime, we find an asymptotic Gaussian law. In the critical-index regime, we also find an asymptotic Gaussian law, albeit a difference in the scale factor, which involves logarithmic terms. In both of these regimes, we have explicit forms for the structure of the mean and the covariance matrix of the composition vector (both exact and asymptotic). In all three regimes we have strong laws.

Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of~$q$-enumeration (this is a model where objects with a parameter of value $k$ have a Gibbs measure/Boltzmann weight $q^k$). For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value $q=q_c$, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime ($q>q_c$), and it is a Boltzmann distribution in the subcritical regime ($0

We consider the following card guessing game with no feedback. An ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards. One after another a single card is drawn from the top, the guesser makes a guess without seeing the card and gets no response if the guess was correct or not. Building upon and improving earlier results, we provide a limit law for the number of correct guesses and also show convergence of the integer moments., Comment: 18 pages

We consider a card guessing game with complete feedback. A ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards, where one after another a single card is drawn from the top, and shown to the guesser until no cards remain. Improving earlier results, we provide a limit law for the number of correct guesses. As a byproduct, we relate the number of correct guesses in this card guessing game to the number of correct guesses under a two-color card guessing game with complete feedback. Using this connection to two-color card guessing, we can also show a limiting distribution result for the first occurrence of a pure luck guess., Comment: 18 pages, 2 figures; (references updated in version 2)

We consider a card guessing strategy for a stack of cards with two different types of cards, say $m_1$ cards of type red (heart or diamond) and $m_2$ cards of type black (clubs or spades). Given a deck of $M=m_1+m_2$ cards, we propose a refined counting of the number of correct color guesses, when the guesser is provided with complete information, in other words, when the numbers $m_1$ and $m_2$ and the color of each drawn card are known. We decompose the correct guessed cards into three different types by taking into account the probability of making a correct guess, and provide joint distributional results for the underlying random variables as well as joint limit laws., Comment: 21 pages, 4 figure

This short note serves an addendum to the article "Destruction of very simple trees" by Fill, Kapur and Panholzer (2004). Therein, the limit law of one-sided tree destruction with a toll function was determined by its moment sequence. We add an identification of the limit law, using recent results of Bertoin (2022), in terms of the local time of a noise reinforced Bessel process., Comment: 3 Pages

We study several exactly solvable Polya-Eggenberger urn models with a \emph{diminishing} character, namely, balls of a specified color, say $x$ are completely drawn after a finite number of draws. The main quantity of interest here is the number of balls left when balls of color $x$ are completely removed. We consider several diminishing urns studied previously in the literature such as the pills problem, the cannibal urns and the OK Corral problem, and derive exact and limiting distributions. Our approach is based on solving recurrences via generating functions and partial differential equations., Comment: 11 pages, 2 figures; appeared in the Proceedings of the Formal Power Series and Algebraic Combinatorics (FPSAC 2007) Nankai University, Tianjin, China, 2007 (only printed proceedings). arXiv admin note: text overlap with arXiv:1110.2425

We provide a fundamental result for bucket increasing trees, which gives a complete characterization of all families of bucket increasing trees that can be generated by a tree evolution process. We also provide several equivalent properties, complementing and extending earlier results for ordinary increasing trees to bucket trees. Additionally, we state second order results for the number of descendants of label $j$, again extending earlier results in the literature., Comment: Reference updated; 22 pages; draws upon and solves an open problem of arXiv:2003.06450

Multitudinous combinatorial structures are counted by generating functions satisfying a composition scheme $F(z)=G(H(z))$. The corresponding asymptotic analysis becomes challenging when this scheme is critical (i.e., $G$ and $H$ are simultaneously singular). The singular exponents appearing in the Puiseux expansions of $G$ and $H$ then dictate the asymptotics. In this work, we first complement results of Flajolet et al. for a full family of singular exponents of $G$ and $H$. Motivated by many examples (random mappings, planar maps, directed lattice paths), we consider a natural extension of this scheme, namely $F(z,u)=G(u H(z))M(z)$. We also consider a variant of this scheme, which allows us to analyse the number of $H$-components of a given size in $F$. These two models lead to a rich world of limit laws, where we identify the key r\^ole played by a new universal three-parameter law: the beta-Mittag-Leffler distribution, which is essentially the product of a beta and a Mittag-Leffler distribution. We prove (double) phase transitions, additionally involving Boltzmann and mixed Poisson distributions, with a unified explanation of the associated thresholds. We also obtain moment convergence and local limit theorems. We end with extensions of the critical composition scheme to a cycle scheme and to the multivariate case, leading to product distributions. Applications are presented for random walks, trees (supertrees of trees, increasingly labelled trees, preferential attachment trees), triangular P\'olya urns, and the Chinese restaurant process., Comment: 57 pages; dedicated to Alois Panholzer on the occasion of his 50th birthday. This version 2 is strongly enhanced with new theorems, applications, more references, etc

In this note we introduce multi-interpolated multiple zeta values. We provide a basic decomposition of these objects involving ordered partitions. We also obtain identities for special instances of multi-interpolated multiple zeta values $\zeta^{\vec{t}}(\{s\}_k)$, generalizing earlier results. Moreover, we introduce a product for multi-interpolated multiple zeta values., Comment: 18 pages; contains material from an old version 3 of arXiv:1903.07346, which has been split

In this work we analyze bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size $n$, complementing the earlier result of Mahmoud and Smythe for bucket recursive trees. On the combinatorial side, we define multilabelled generalizations of the tree families $d$-ary increasing trees and generalized plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al., Comment: 28 pages (5 figures, 2 tables, 4 algorithms)

We study logarithmic integrals of the form $\int_0^1 x^i\ln^n(x)\ln^m(1-x)dx$. They are expressed as a rational linear combination of certain rational numbers $(n,m)_i$, which we call tiered binomial coefficients, and products of the zeta values $\zeta(2)$, $\zeta(3)$,\dots. Various properties of the tiered binomial coefficients are established. They involve, amongst others, the binomial transform, truncated multiple zeta and multiple zeta star values, as well as special functions. As an application we discuss the limit law of the number of comparisons of the Quicksort algorithm: we reprove that the moments of the limit law are rational polynomials in the zeta values. A novel expression for the cumulants of the Quicksort limit is also presented., Comment: 31 pages

In this work we discuss a parameter $\sigma$ on weighted $k$-element multisets of $[n]= \{1,\dots ,n\}$. The sums of weighted $k$-multisets are related to $k$-subsets, $k$-multisets, as well as special instances of truncated interpolated multiple zeta values. We study properties of this parameter using symbolic combinatorics. We rederive and extend certain identities for $\zeta^{t}_n(\{m\}_k)$. Moreover, we introduce random variables on the $k$-element multisets and derive their distributions, as well as limit laws for $k$ or $n$ tending to infinity., Comment: 24 pages, no figure; final version to appear in Electronic Journal of Combinatorics

We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers., Comment: 6 pages

We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures and give an analysis of some parameters., Comment: Journal version of arXiv:1605.02307

We obtain an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1{n}}dx$ as $n\to\infty$, and compute $I_j$ in terms of alternating (or "colored") multiple zeta value. We also show that $I_j$ is a rational polynomial the ordinary zeta values, and give explicit formulas for $j\le 12$. As a byproduct, we obtain precise results about the convergence of norms of random variables and their moments. We study $\Vert(U,1-U)\Vert_n$ as $n$ tends to infinity and we also discuss $\Vert(U_1,U_2,\dots,U_r)\Vert_n$ for standard uniformly distributed random variables., Comment: 27 pages

We consider multicolor urn models with multiple drawings. An urn model is called linear if the conditional expected value of the urn composition at time $n$ is a linear function of the composition at time $n-1$. For four different sampling schemes - ordered and unordered samples with or without replacement - we classify urns into linear and non-linear models. We also discuss representations of the expected value and the covariance for linear models., Comment: 13 pages. This is a preliminary version. Comments welcome!

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths., Comment: One of the proceedings-papers of the conference AofA 2016: 27th International conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Krakow, Poland, July 4-8, 2016

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $x \oplus y$, is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions., Comment: 31 pages, 2 figures

In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits., Comment: 17 pages

This is the second part of a two-part investigation. We continue the study of a class of balanced urn schemes on balls of two colors (white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn and ball addition rules are applied; the special case $m=1$ of sampling only a single ball coincides with ordinary balanced urn models. We consider these multiple drawings under sampling with or without replacement. For the class of affine conditional expected value, we study the number of white balls after $n$ steps. The affine class is parametrized by $\Lambda$, specified by the ratio of the two eigenvalues of a reduced ball replacement matrix and the sample size, leading to three different cases: small-index urns ($\Lambda \le \frac 1 2$, and the case $\Lambda= \frac 1 2$ is critical), large-index urns ($\Lambda > \frac 1 2$), and triangular urns. In Part I we derived a central limit theorem for small index urns, and proved almost-sure convergence for large index and triangular urn models. In the present paper (Part II), we continue the study of affiance urn schemes and study the moments of large-index urns and triangular urn models. We show moment convergence under suitable scaling and we also provide expressions for the moments., Comment: 22 pages

This is a research endeavor in two parts. We study a class of balanced urn schemes on balls of two colours (say white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn, and ball addition rules are applied. We consider these multiple drawings under sampling with or without replacement. We further classify ball addition matrices according to the structure of the expected value into affine and nonaffine classes. We give a necessary and sufficient condition for a scheme to be in the affine subclass. For the affine subclass, we get explicit results for the expected value and second moment of the number of white balls after $n$ steps and an asymptotic expansion of the variance. Moreover, we uncover a martingale structure, amenable to a central limit theorem formulation. This unifies several earlier works focused on special cases of urn models with multiple drawings. The class is parametrized by $\Lambda$, specified by the ratio of the two eigenvalues of a "reduced" ball replacement matrix and the sample size. We categorize the class into small-index urns ($\Lambda <\frac 1 2$), critical-index urns ($\Lambda = \frac 1 2$), and large-index urns ($\Lambda > \frac 1 2$), and triangular urns. In the present paper (Part I), we obtain central limit theorems for small- and critical-index urns and prove almost-sure convergence for triangular and large-index urns. In a companion paper (Part II), we discuss the moment structure of large-index urns and triangular urns., Comment: 26 pages

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree families of an elliptic nature, where the exponential generating function can be expressed in terms of the Weierstrass-p function or the lemniscate sine function. Furthermore, we show how to translate enumeration formulas for multilabelled increasing trees into hook-length formulas for trees and present a general "reverse engineering" method to discover hook-length formulas associated to such tree families., Comment: 37 pages

In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable $X$ with moment sequence $(\mu_s)_{s\in\mathbb{N}}$ we determine a discrete random variable $Y$, whose moment sequence is given by the Stirling transform of the sequence $(\mu_s)_{s\in\mathbb{N}}$, and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a simple limit theorem based on expansions of factorial moments instead of power moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyse triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time $n$. We discuss the branching structure of plane recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley-trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed Poisson-Rayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics., Comment: 61 pages, 13 figures

We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate $\ell$ selected nodes in a size-n random recursive tree for three different selection rules, namely (i) isolating all of the nodes labelled 1,2,...,$\ell$ (thus nodes located close to the root of the tree), (ii) isolating all of the nodes labelled n+1-$\ell$,n+2-$\ell$,...,n (thus nodes located at the fringe of the tree), and (iii) isolating $\ell$ nodes in the tree, which are selected at random before starting the edge-removal procedure. Using a generating functions approach we determine for these selection rules the limiting distribution behavior of the number of cuts to isolate all selected nodes, for $\ell$ fixed and n tending to infinity., Comment: 25 pages, 2 figures

We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem, originally introduced by Knuth and McCarthy, and generalizations of the well known sampling without replacement urn models, and OK Corral urn models., Comment: 12 Pages, No figure

We study an urn model introduced in the paper of Chen and Wei, where at each discrete time step $m$ balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known P\'olya-Eggenberger urn model, case m=1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model., Comment: 15 pages, no figures

In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees., Comment: 4 pages.

In this work we discuss two urn models with general weight sequences $(A,B)$ associated to them, $A=(\alpha_n)_{n\in\N}$ and $B=(\beta_m)_{m\in\N}$, generalizing two well known P\'olya-Eggenberger urn models, namely the so-called sampling without replacement urn model and the OK Corral urn model. We derive simple explicit expressions for the distribution of the number of white balls, when all black have been drawn, and obtain as a byproduct the corresponding results for the P\'olya-Eggenberger urn models. Moreover, we show that the sampling without replacement urn models and the OK Corral urn models with general weights are dual to each other in a certain sense. We discuss extensions to higher dimensional sampling without replacement and OK Corral urn models, respectively, where we also obtain explicit results for the probability mass functions, and also an analog of the dualitiy relation. Finally, we derive limit laws for a special choice of the weight sequences., Comment: 19 pages, no figures

A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutations). We establish a bijection of these local types to node types of $(k+1)$-ary increasing trees. We present a branched continued fraction representation of the generating function of these local types through a bijection with path diagrams induced by \L ukasiewicz paths, generalizing the results from permutations to arbitrary $k$-Stirling permutations. We further show that the generating function of ordinary Stirling permutation has at least three branched continued fraction representations, using correspondences between non-standard increasing trees, $k$-Stirling permutations and path diagrams., Comment: 28 pages

Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded $d$-ary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice paths can be solved without using the kernel method by regarding simple families of lattice paths as degenerated families of embedded trees., Comment: 19 pages, 2 Figures; typos corrected; Section 8 added

It was shown in that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from \cite{KirProd98,ProSchnKu} can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a simple elementary proof of the shuffle identity using only partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums., Comment: 8 pages

We study for $s\in\N=\{1,2,...\}$ the functions $\xi_{k}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}\Li_{k}(1-e^{-t})dt$, and more generally $\xi_{k_1,...,k_r}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}\Li_{k_1,...,k_r}(1-e^{-t})dt$, introduced by Arakawa and Kaneko \cite{Arakawa} and relate them with (finite) multiple zeta functions, partially answering a question of \cite{Arakawa}. In particular, we give an alternative proof of a result of Ohno \cite{Ohno2}., Comment: 5 pages

In this note we consider ternary trees naturally embedded in the plane in a deterministic way such that the root has position zero, or in other words label zero, and the children of a node with position $j$ have positions $j-1$, $j$, and $j+1$, for all $j\in\Z$. We derive the generating function of ternary trees where all nodes have labels which are less or equal than $j$, with $j\in\N$, and the generating function of ternary trees counted with respect to nodes with label $j$, with $j\in\Z$. Moreover, we discuss generalizations of the counting problem to several labels at the same time. Furthermore, we use generating functions to study the depths of the external node $s$, or in other words leaf $s$ with $0\le s\le 2n$, where the $2n+1$ external nodes of a ternary tree are numbered from the left to the right according to an inorder traveral. The three different types depths -- left, right and center -- are due to the embedding of the ternary tree in the plane. Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees., Comment: 15 pages, 5 figures; Version 2: typos corrected, simplified formula for series $X$ added

Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods., Comment: 25 pages, 3 figures, Keywords: Increasing trees, plane recursive trees, Stirling permutations, ascents, descents, urn models, limiting distributions