Partitions of the set of finite sequences

AbstractWe consider partitions of the set of finite sequences of a cardinal and investigate the existence of different kinds of homogeneous objects for them. We prove some properties of the first cardinal satisfying a natural partition relation and study how different types of homogeneitics are interrelated

Two New Integer Sequences Related to Crossroads and Catalan Numbers

The lonely singles sequence represents the number of noncrossing partitions of the finite set {1,. .. , n} in which no pair of singletons {i} and {j} can be merged into the pair {i, j} so that the partition stays noncrossing. The marriageable singles sequence represents the number of all the other noncrossing partitions and is the difference between the Catalan numbers sequence and the lonely singles sequence. The 14 first terms of these sequences are given, as well as some of their properties. These sequences appear when one wants to count the number of ways to cross simultaneously certain road intersections.Comment: Submitted to the Journal of Integer Sequences on November 11, 202

On the order sequence of a group

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. Abelian groups are uniquely determined by their order sequences, and the poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.Comment: 22 pages, Comments are most welcom

Cantor's theorem may fail for finitary partitions

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite set $A$ and a surjection from $A$ onto $\mathscr{B}(A)$. On the other hand, we prove in $\mathsf{ZF}$ some theorems concerning $\mathscr{B}(A)$ for infinite sets $A$, among which are the following: (1) If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$ and no finite-to-one functions from $\mathscr{B}(A)$ to $A$. (2) For all $n\in\omega$, $|A^n|<|\mathscr{B}(A)|$. (3) $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$, where $\mathrm{seq}(A)$ is the set of all finite sequences of elements of $A$.Comment: 19 page

Coding Partitions of Regular Sets *

Abstract A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the factorizations of finite sequences. We prove that the canonical coding partition of a regular set contains a finite number of regular classes and we give an algorithm for computing this partition. From this we derive a canonical decomposition of a regular monoid into a free product of finitely many regular monoids

Deterministic stack-sorting for set partitions

A sock sequence is a sequence of elements, which we will refer to as socks, from a finite alphabet. A sock sequence is sorted if all occurrences of a sock appear consecutively. We define equivalence classes of sock sequences called sock patterns, which are in bijection with set partitions. The notion of stack-sorting for set partitions was originally introduced by Defant and Kravitz. In this paper, we define a new deterministic stack-sorting map $\phi_{\sigma}$ for sock sequences that uses a $\sigma$-avoiding stack, where pattern containment need not be consecutive. When $\sigma = aba$, we show that our stack-sorting map sorts any sock sequence with $n$ distinct socks in at most $n$ iterations, and that this bound is tight for $n \geq 3$. We obtain a fine-grained enumeration of the number of sock patterns of length $n$ on $r$ distinct socks that are $1$-stack-sortable under $\phi_{aba}$, and we also obtain asymptotics for the number of sock patterns of length $n$ that are $1$-stack-sortable under $\phi_{aba}$. Finally, we show that for all unsorted sock patterns $\sigma \neq a\cdots a b a \cdots a$, the map $\phi_{\sigma}$ cannot eventually sort all sock sequences on any multiset $M$ unless every sock sequence on $M$ is already sorted