Related papers: Juggling card sequences
This paper explores the number of parallelograms that appear in a billiard path that enters one corner of a rectangle and leaves a second corner of a rectangle as a function of the normalized dimensions of the rectangle.
We prove some partial results on the periodicity of billiard systems on graphs. The results specialize to the case of $n$ billiards with equal mass on the unit interval or circle traveling at the same speed.
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…
Choreographies describe possible sequences of interactions among a set of agents. We aim to join two lines of research on choreographies: the use of the shuffle on trajectories operator to design more expressive choreographic languages, and…
Consider n cards that are labeled 1 through n with n an even integer. The cards are put face down and their ordering starts with card labeled 1 on top through card labeled n at the bottom. The cards are top to random shuffled m times and…
Have you ever played or watched a game of pool? If so, you have already seen a billiard system in action. In mathematics and physics, a billiard system describes a ball that moves in straight lines and bounces off walls. Despite these…
Biclustering is a powerful approach to search for patterns in data, as it can be driven by a function that measures the quality of diverse types of patterns of interest. However, due to its computational complexity, the exploration of the…
We study properties of an array of numbers, called "the triangle," in which each row is formed by rotating all the numbers in the previous row to the left by $m$ positions in cyclical fashion, then appending a number to the end of the row.…
We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
A discrete map based on the sum of an integer's distinct primes factors and the sum of its other factors is defined and its iteration is studied.
This paper is about the following question: How many riffle shuffles mix a deck of card for games such as blackjack and bridge? An object that comes up in answering this question is the descent polynomial associated with pairs of decks,…
Three events in a probability space form a conjunctive fork if they satisfy specific constraints on conditional independence and covariances. Patterns of conjunctive forks within collections of events are characterized by means of systems…
A deck of $n$ cards is shuffled by repeatedly moving the top card to one of the bottom $k_n$ positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as $k_n$ ranges from a constant…
A mixed graph can be seen as a type of digraph containing some edges (two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures…
We develop a combinatorial and order-theoretic framework for shuffles, understood as ordered concatenations of indexed families of sequences that induce total orders on the natural numbers. Motivated by the classical \v{S}arkovski\u{i}…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
The preference graph is a combinatorial representation of the structure of a normal-form game. Its nodes are the strategy profiles, with an arc between profiles if they differ in the strategy of a single player, where the orientation…
This paper discusses the formalization of proofs "by diagram chasing", a standard technique for proving properties in abelian categories. We discuss how the essence of diagram chases can be captured by a simple many-sorted first-order…
We study the functorial and growth properties of closed orbits for maps. By viewing an arbitrary sequence as the orbit-counting function for a map, iterates and Cartesian products of maps define new transformations between integer…