Related papers: Deterministic stack-sorting for set partitions
Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the…
Let pi = pi_1 pi_2 ... pi_n be a permutation in the symmetric group S_n written in one-line notation. The pinnacle set of pi, denoted Pin pi, is the set of all pi_i such that pi_{i-1} < pi_i > pi_{i+1}. This is an analogue of the…
Sequence theories are an extension of theories of strings with an infinite alphabet of letters, together with a corresponding alphabet theory (e.g. linear integer arithmetic). Sequences are natural abstractions of extendable arrays, which…
In this paper we introduce a family of partitions of the set of natural numbers, Fibonacci-like partitions. In particular, we introduce a Fibonacci-like partition in a number of parts corresponding to the Fibonacci numbers, the standard…
We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let $\op_{n,k}(p)$ be the number of ordered set partitions of $\{1,2,\ldots,n\}$ into $k$ blocks that…
The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear domain, and we can…
The signature of a path, introduced by K.T. Chen [5] in $1954$, has been extensively studied in recent years. The $2010$ paper [12] of Hambly and Lyons showed that the signature is injective on the space of continuous finite-variation paths…
This paper introduces a new comparison base stable sorting algorithm, named RS sort. RS Sort involves only the comparison of pair of elements in an array which ultimately sorts the array and does not involve the comparison of each element…
In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely specify the stack. Our main result shows that…
We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof…
Object detection or localization is an incremental step in progression from coarse to fine digital image inference. It not only provides the classes of the image objects, but also provides the location of the image objects which have been…
Sorting an array is a fundamental routine in machine learning, one that is used to compute rank-based statistics, cumulative distribution functions (CDFs), quantiles, or to select closest neighbors and labels. The sorting function is…
A tree $T$ on $2^n$ vertices is called set-sequential if the elements in $V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional $01$-vectors such that the vector labeling each edge is the component-wise sum modulo $2$ of…
A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly…
Shapley and Scarf introduced a notion of stable allocation between traders and indivisible goods, when each trader has rank-ordered each of the goods. The purpose of this note is to prove that the distribution of ranks after allocation is…
Point pattern sets arise in many different areas of physical, biological, and applied research, representing many random realizations of underlying pattern formation mechanisms. These pattern sets can be heterogeneous with respect to…
We consider a new class of matrices associated to a real square matrix $A$ and to a vector $\vec{c} \in \{-1,1\}^n$ such that $c_1=1$ by using a map $\varphi_{\vec{c}}$ which turns out to be a conjugation of a matrix $A$ by a signature…
This paper introduces a class of objects called decision rules that map infinite sequences of alternatives to a decision space. These objects can be used to model situations where a decision maker encounters alternatives in a sequence such…
Conformal prediction is a statistical framework that generates prediction sets containing ground-truth labels with a desired coverage guarantee. The predicted probabilities produced by machine learning models are generally miscalibrated,…
The order and chain polytopes, introduced by Richard P. Stanley, form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li states that the $f$-vector of the chain polytope…