Banach's matchbox problem considers the setting of two matchboxes that each initially contain the same number of matches. Boxes are chosen with equal probability and a match removed each time. The problem concerns the law of the number of matches remaining in one box once the other box empties. Knuth considered a generalization of this problem whereby big-choosers arrive with probability pand remove a match from the box with the most number remaining, and little-choosers arrive with probability 1-p and remove a match from the box with the least number remaining. In this paper we consider Knuth's generalization for the case of kmatchboxes in which there are big-choosers and little-choosers. We determine the generating function for the expected number of matches remaining in k-1 matchboxes once a box first empties, a quantity we refer to as the 'residue'. Interestingly, this generating function is a quotient whose denominator contains a generating function for a special case of the Raney numbers. The form for this generating function allows us to give an expression for the expected residue in terms of a sum that involves diagonal state return probabilities, where a diagonal state is a configuration in which all matchboxes each contain the same number of matches. We use analytic techniques to determine the asymptotic behavior of this expected value for all values of p, which involves the study of an asymmetric random walk. In addition to this we consider the expected value of the order of the first return to a diagonal state and determine the asymptotic behavior of this quantity. The coefficients of the diagonal state probability generating function are shown to be related to 'manila folder configurations in a filing cabinet', and we make this connection precise. This allows us to use known results for the enumeration of such manila folder configurations to give a closed form expression for the diagonal state return probabilities. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially the generating function of the level statistic on the set of recurrent configurations, called stochastically recurrent states, for that model. In this note we consider the lacking polynomial of the complete bipartite graph. We classify the stochastically recurrent states of the stochastic sandpile model on the complete bipartite graphs K2,n and Km,2 where the sink is always an element of the set counted by the first index. We use these characterizations to give explicit formulae for the lacking polynomials of these graphs. Log-concavity of the sequence of coefficients of these two lacking polynomials is proven, and we conjecture log-concavity holds for this general class of graphs.

In this paper we consider a general system of activities that can, but do not have to, occur. This system is governed by a set of two types of constraints: precedence and response. A precedence constraint dictates that an activity can only occur if it has been preceded by some other specified activity. Response constraints are similarly defined. An execution of the system is a listing of activities in the order they occur and which satisfies all constraints. Such systems naturally arise in areas of theoretical computer science and decision science. An outcome of the freedom with which activities can occur is that there are many different possible executions, and gaining a combinatorial insight into these is a non-trivial problem. We characterize all of the ways in which such a system can be executed. Our approach uses order theory to provide a classification in terms of the linear extensions of posets constructed from the constraint sets. This characterization is essential in calculating the stakeholder utility metrics that have been developed by the first author that allow for quantitative comparisons of such systems/processes. It also allows for a better understanding of the theoretical backbone to these processes and their deconstruction as a shuffle product of smaller systems.

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs G_μ ,ν , which we call clique-independent graphs, indexed by two compositions μ and ν . Moreover, we define a delay statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the celebrated shuffle theorem of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials ⟨∇ e_n, e_μ h_ν⟩ in terms of these configurations.

In this paper, we investigate a permutation statistic that was independently introduced by Romik (Funct Anal Appl 39(2):52–155, 2005). This statistic counts the number of bumps that occur during the execution of the Robinson–Schensted procedure when applied to a given permutation. We provide several interpretations of this bump statistic that include the tableaux shape and also as an extremal problem concerning permutations and increasing subsequences. Several aspects of this bump statistic are investigated from both structural and enumerative viewpoints.

This paper studies sorted recurrent configurations of the Abelian sandpile model on the complete split graph. We introduce two natural toppling processes, CTI and ITC toppling, on the recurrent configurations and use these to define two toppling delay statistics, wtopple_CTI and wtopple_ITC. These new toppling delay statistics are time-weighted sums for the number of vertices that topple during each iteration of the toppling processes. We then introduce the bivariate q,t-CTI and q,t-ITC polynomials that are the generating functions of the bistatistics (level,wtopple_ITC) and (level,wtopple_CTI), where level is the well-established sandpile level statistic. We prove the bistatistic (level,wtopple_ITC) maps to a bistatistic (area,bounce) on Schröder paths that was introduced by Egge, Haglund, Killpatrick and Kremer (2003). This establishes equality of the q,t-ITC polynomial and the q,t-Schröder polynomial of those same authors. This connection allows us to relate the q,t-ITC polynomial to the theory of symmetric functions and also establishes symmetry of the q,t-ITC polynomials. We conjecture equality of the q,t-CTI and q,t-ITC polynomials. We also present and prove a characterization of sorted recurrent configurations as a new class of polyominoes that we call sawtooth polyominoes. The CTI and ITC toppling processes on sorted recurrent configurations are proven to correspond to bounce paths within the polyominoes. The main difference between the two bounce paths is the initial direction in which they travel. In addition to this, and building on the results of Aval, D'Adderio, Dukes, and Le Borgne (2016), we present a cycle lemma for a slight extension of stable configurations that allows for an enumeration of sorted recurrent configurations within the framework of the sandpile model.

Let alpha = a_1 a_2 ... a_n be a sequence of nonnegative integers. The ascent set of alpha, Asc(alpha), consists of all indices k where a_{k+1} > a_k. An ascent sequence is alpha where the growth of the a_k is bounded by the elements of Asc(alpha). These sequences were introduced by Bousquet-M\'elou, Claesson, Dukes and Kitaev and have many wonderful properties. In particular, they are in bijection with unlabeled (2+2)-free posets, permutations avoiding a particular bivincular pattern, certain upper-triangular nonnegative integer matrices, and a class of matchings. A weak ascent of alpha is an index k with a_{k+1} >= a_k and weak ascent sequences are defined analogously to ascent sequences. These were studied by B\'enyi, Claesson and Dukes and shown to have analogous equinumerous sets. Given a nonnegative integer d, we define a difference d ascent to be an index k such that a_{k+1} > a_k - d. We study the properties of the corresponding d-ascent sequences, showing that some of the maps from the weak case can be extended to bijections for general d while the extensions of others continue to be injective (but not surjective). We also make connections with other combinatorial objects such as rooted duplication trees and restricted growth functions.

The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with $n$ vertices is more than 100 years old (MacMahon 1920). In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences.

In this paper we introduce weak ascent sequences, a class of number sequences that properly contains ascent sequences. We show how these sequences uniquely encode each of the following objects: permutations avoiding a particular length-4 bivincular pattern; upper-triangular binary matrices that satisfy a column-adjacency rule; factorial posets that are weakly (3+1)-free. We also show how weak ascent sequences are related to a class of pattern avoiding inversion sequences that has been a topic of recent research by Auli and Elizalde. Finally, we consider the problem of enumerating these new sequences and give a closed form expression for the number of weak ascent sequences having a prescribed length and number of weak ascents.

In this paper we will provide a quantitative analysis of a simple model of the Federal Disaster Assistance policy from the viewpoint of three different stakeholders. This quantitative methodology is new and has applications to other areas such as business and healthcare processes. The stakeholders are interested in process transparency but each has a different opinion on precisely what constitutes transparency. We will also consider three modifications to the Federal Disaster Assistance policy and analyse, from a stakeholder viewpoint, how stakeholder satisfaction changes from process to process. This analysis is used to rank the favourability of four policies with respect to all collective stakeholder preferences.

In this paper we define a variant of the Ising model in which spins are replaced with permutations. The energy between two spins is a function of the relative disorder of one spin, a permutation, to the other. This model is motivated by a complexity measure for declarative systems. For such systems a state is a permutation and the permutation sorting complexity measures the average sequential disorder of neighbouring states. To measure the relative disorder between two spins we use a symmetrized version of the descent permutation statistic that has appeared in the works of Chatterjee & Diaconis and Petersen. The classical Ising model corresponds to the length-2 permutation case of this new model. We consider and prove some elementary properties for the 1D case of this model in which spins are length-3 permutations.

In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.

We present a method for calculating and analyzing stakeholder utilities of processes that arise in, but are not limited to, social sciences. These areas include business process analysis, healthcare workflow analysis, and policy process analysis. This method is quite general and applicable to any situation in which declarative-type constraints of a modal and/or temporal nature play a part. A declarative process is a process in which activities may freely happen while respecting a set of constraints. For such a process, anything may happen so long as it is not explicitly forbidden. Declarative processes have been used and studied as models of business and healthcare workflows by several authors. In considering a declarative process as a model of some system, it is natural to consider how the process behaves with respect to stakeholders. We derive a measure for stakeholder utility that can be applied in a very general setting. This derivation is achieved by listing a collection of properties that we argue such a stakeholder utility function ought to satisfy and then using these to show that a very specific form must hold for such a utility. The utility measure depends on the set of unique traces of the declarative process, and calculating this set requires a combinatorial analysis of the declarative graph that represents the process. This builds on previous work of Dukes and Casey (2021) in which the combinatorial diversity metrics for declarative processes were derived for use in policy process analysis. The collection of stakeholder utilities can themselves then be used to form a metric with which we can compare different declarative processes to one another. These are illustrated using several examples of declarative processes that already exist in the literature.

In this paper we examine a procedure that, on starting with an integer n, results in a pair of equal integers that are no greater than n. We call the resulting value the strange root of n and we show how this strange-root-finding procedure is intimately linked to the game of Tchoukaillon solitaire. We analyze the strange-root-finding procedure in reverse to determine when a prescribed value is the strange root of at most two integers. We present a conjecture about strange roots and translate this conjecture into one involving Tchoukaillon solitaire.

We highlight some results from studying chip-firing on the the complete split graph [5]. In this work it is shown that recurrent states can be characterised in terms of Motzkin words and can also be characterised in terms of a new type of parking function that we call a tiered parking function. These new parking functions arise by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them.

We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them. We also enumerate the different sets of recurrent configurations studied in this paper, and in doing so derive a formula for the number of spanning trees of the complete split graph that uses a bijective Prüfer code argument.

We present several completely general diversity metrics to quantify the problem-solving capacity of any public policy decision-making process. These metrics can be used alongside existing policy process analysis methods and capture aspects of the policy process that have not been considered before. This is performed by modelling the policy process using a declarative process paradigm. We introduce a class of traces, called first-passage traces, to represent different executions of the declarative processes. Heuristics of what properties a diversity measure of such processes ought to satisfy are used to derive two different metrics for these processes in terms of the set of first-passage traces. These metrics turn out to have formulations in terms of the entropies of two different random variables on the set of traces of the processes. In addition, we introduce a measure of “goodness” that allows for comparisons of policy processes with respect to the prescribed notion of “goodness”.

In this paper we examine a procedure (due to Colm Fagan) that, on starting with an integer n, results in a pair of equal integers that are no greater than n. We call the resulting value the strange root of n and we show how this strange root finding procedure is intimately linked to the game of Tchoukaillon solitaire. We analyse the strange root finding procedure in reverse to be able to give a method for determining when a prescribed value is the strange root of at most two integers.

This paper establishes connections between EW-tableaux and parallelogram polyominoes by using recent research regarding the sandpile model on the complete bipartite graph. This paper presents and proves a direct bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes. The significance of this is that allows one to move between these objects without the need for `recurrent configurations', the central object which previously tied this work together. It introduces the notion of a marked rectangular EW-tableaux that exactly encode all recurrent configurations of the sandpile model on the complete bipartite graph. This shows how non-cornersupport entries that featured in previous work can be utilized in a simple but important way in relation to EW-tableaux. It lifts the bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes to a bijection between marked rectangular EW-tableaux and labelled parallelogram polyominoes. This bijection helps us to fully understand the aspects of these very different objects that are, in a sense, different sides of the same coin.

A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs.