arXiv subjects

Markus Kuba

Publications and source records attributed to Markus Kuba.

At least 19 recordsLinked to original sources

arXiv2026

On card guessing after an asymmetric single-shelf shuffle

We provide a definitive analysis of the number of correct guesses in the complete-feedback card guessing game after an asymmetric single-shelf shuffle with parameter $p\in (0, 1)$. We explicitly describe the optimal strategy that maximizes the expected number of correct guesses. We study the number of correct guesses, under an optimal strategy, using methods from analytic combinatorics. In addition, we find the explicit distribution for the number of correct guesses, and thus find the mean and the variance for the number of correct guesses. We show that the distribution of the number of correct guesses is log-concave. We also study the limiting behaviour of the number of correct guesses as the number of cards goes to infinity. In particular, we prove a (local) central limit theorem and a large deviation principle with an explicit rate function. We also prove phase transitions for the number of correct guesses near $p=0$ and $p=1$. Prior to this work, only the optimal strategy and the expected number of correct guesses under the optimal strategy were known for the special case of $p=1/2$.

math.CO↗

arXiv2026

Asymptotic expansions of integrals and Nielsen's polylogarithms

This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the coefficients of the asymptotic expansions to Nielsen's generalized polylogarithms. For $q=-1$, we obtain an expansion in terms of multiple zeta values, which in this setting, reduce to ordinary zeta values. A key point is that for $q=1$, the integrals typically produce alternating multiple zeta values; we formulate a precise symmetry constraint on the relevant coefficient sequence under which all coefficients reduce to polynomials in ordinary zeta values. We also translate this symmetry into a statement about a binomial transform, and we verify the condition for several classical Appell-type families, like Euler, Bernoulli, Genocchi, and Hermite. Finally, we obtain precise results about the convergence of norms of random variables.

math.NT↗

arXiv2026

On Card guessing after a single shelf shuffle

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is shelf-shuffled exactly one time. One after the other a single card is drawn from the shuffled deck. The guesser makes has guess and the card is shown until no cards remain. We provide a distributional analysis of the number of correct guesses under the optimal strategy. We re-obtain the previously derived expectation and add a complete description of the distribution. We also obtain a central limit theorem for the number $n$ of cards tending to infinity. Furthermore, we discuss an unbalanced, biased shelf shuffle and show how to derive the extend our analysis, also adding the complete position matrix. Finally, a refined analysis of the number of correct guesses is carried out, distinguishing between pure luck guesses and certified correct guesses.

math.CO↗

arXiv2026

Card guessing after an asymmetric riffle shuffle

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is riffle-shuffled exactly one time. Given a value $p\in(0{,}1)\setminus\{\frac12\}$, the riffle shuffle is assumed to be unbalanced, such that the cut is expected to happen at position $p\cdot n$. 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, and shown to the guesser until no cards remain. We provide a detailed analysis of the optimal guessing strategy and study the distribution of the number of correct guesses.

math.CO↗

arXiv2025

A note on mixed Poisson distributions

In this note we discuss additional properties of mixed Poisson distributions. We discuss the convergence of mixed Poisson distributions to its mixing distribution for the scaling parameter tending to infinity. Moreover, we obtain a central limit theorem after centering by its mixing random variable, together with moment convergence.

math.PR↗

arXiv2024

Gibbs partitions and lattice paths

This work is devoted to the analysis of a Gibbs partition model, also known as a composition scheme. We consider a natural new condition on the component weights. It leads to a new behavior for the total number of components. We discover a condensation phenomenon, producing a unique giant component comprising almost the entire mass. Additionally, we prove a point process limit describing the asymptotic size of the non-maximal components exhibiting a sublinear power-law growth. A particular motivation for our article stems from applications, ranging from simple random walks in the cube, over lattice paths models in the plane, pairs of directed random walks, over to urn models and card guessing games.

math.CO↗

arXiv2024

On P\'olya-Young urn models and growth processes

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.

math.PR↗

arXiv2024

On affine multi-color urns grown under multiple drawing

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.

math.PR↗

arXiv2023

Composition schemes: q-enumerations and phase transitions

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 ($0<q<q_c$). We apply our results to fundamental statistics of lattice paths and quarter-plane walks. We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons.

math.CO↗

arXiv2023

On Card guessing games: limit law for no feedback one-time riffle shuffle

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.

math.CO↗

arXiv2023

On Card guessing games: limit law for one-time riffle shuffle

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.

math.CO↗

arXiv2023

On Card guessing with two types of cards

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.

math.CO↗

arXiv2023

A note on the limit law of one-sided tree destruction

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.

math.PR↗

arXiv2022

Analysis of some exactly solvable diminishing urn models

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.

math.CO↗

arXiv2022

Tree evolution processes for bucket increasing trees

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.

math.CO↗

arXiv2021

Phase transitions of composition schemes: Mittag-Leffler and mixed Poisson distributions

Multitudinous probabilistic and combinatorial objects are associated with generating functions satisfying a composition scheme $F(z)=G(H(z))$. The analysis becomes challenging when this scheme is critical (i.e., $G$ and $H$ are simultaneously singular). 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 $F$. We prove that these two models lead to a rich world of limit laws, where we identify the key role played by a new universal law introduced in this article: the three-parameter Mittag-Leffler distribution, which is essentially the product of a beta and a Mittag-Leffler distribution. We also prove (double) phase transitions, additionally involving Boltzmann and mixed Poisson distributions, bringing a unified explanation of the associated thresholds. In all cases we 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.

math.PR↗

arXiv2020

On multi-interpolated multiple zeta values

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.

math.CO↗

arXiv2020

On bucket increasing trees, clustered increasing trees and increasing diamonds

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.

math.CO↗