Advances in the Theory of Planar Curve Cognates

Cognate linkages provide the useful property in mechanism design of having the same motion. This paper describes an approach for determining all coupler curve cognates for planar linkages with rotational joints. Although a prior compilation of six-bar cognates due to Dijksman purported to be a complete list, that analysis assumed, without proof, that cognates only arise by permuting link rotations. Our approach eliminates that assumption using arguments concerning the singular foci of the coupler curve to constrain a cognate search and then completing the analysis by solving a precision point problem. This analysis confirms that Dijksman's list for six-bars is comprehensive. As we further demonstrate on an eight-bar and a ten-bar example, the method greatly constrains the set of permutations of link rotations that can possibly lead to cognates, thereby facilitating the discovery of all cognates that arise in that manner. However, for these higher order linkages, the further step of using a precision point test to eliminate the possibility of any other cognates is still beyond our computational capabilities.

Lower bound on the size of a quasirandom forcing set of permutations

A set S of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$ , it holds that $\{\Pi_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer k such that the set of all permutations of order k is forcing; this has been shown to be true for any $k\ge 4$ . In particular, the set of all 24 permutations of order 4 is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.

Simulated annealing method for equilibrium placement problem

The paper proposes a modification of the simulated annealing algorithm as applied to problems that have a fragmented structure. An algorithm for simulating annealing for the traveling salesman problem is considered and its applicability to the optimization problem on a set of permutations is shown. It is proved that the problem of equilibrium placement of point objects on a plane has a fragmentary structure and, therefore, reduces to an optimization problem on a set of permutations. The results of numerical experiments for various types of algorithms for finding the optimal solution in the equilibrium placement problem are presented.

From Permutations to Horizontal Visibility Patterns of Periodic Series

Periodic series of period T can be mapped into the set of permutations of [T−1]={1,2,3,…,T−1}. These permutations of period T can be classified according to the relative ordering of their elements by the horizontal visibility map. We prove that the number of horizontal visibility classes for each period T coincides with the number of triangulations of the polygon of T+1 vertices that, as is well known, is the Catalan number CT−1. We also study the robustness against Gaussian noise of the permutation patterns for each period and show that there are periodic permutations that better resist the increase of the variance of the noise.

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$. We first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by ${\bf B}\circ s$, where ${\bf B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}(\text{Av}_{n,k}(231,312,321))|$, where $\text{Av}_{n,k}(231,312,321)$ is the set of permutations in $\text{Av}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice. Comment: 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1903.09138

JUMPING FROG METHOD FOR OPTIMAL CLASSIFICATIONS

In the article the problem of finding optimal classifications on a finite set is investigated. It is shown that the problem of finding an optimal classification is generated by a tolerance relation on a finite set. It is also reduced to an optimization problem on a set of permutations. It is proposed a modification of the mixed jumping frogs to find suboptimal solutions of the problem of classification.

Permutations Avoiding a Simsun Pattern

A permutation $\pi$ avoids the simsun pattern $\tau$ if $\pi$ avoids the consecutive pattern $\tau$ and the same condition applies to the restriction of $\pi$ to any interval $[k].$ Permutations avoiding the simsun pattern $321$ are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length $3$ in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length $3$ together with any set of classical patterns of length $3.$ The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees.

$4$-Colouring of Generalized Signed Planar Graphs

Assume $G$ is a graph and $S$ is a set of permutations of positive integers. An $S$-signature of $G$ is a pair $(D, \sigma)$, where $D$ is an orientation of $G$ and $\sigma: E(D) \to S$ is a mapping which assigns to each arc $e=(u,v)$ a permutation $\sigma(e)$ in $S$. We say $G$ is $S$-$k$-colourable if for any $S$-signature $(D, \sigma)$ of $G$, there is a mapping $f: V(G) \to [k]$ such that for each arc $e=(u,v)$ of $G$, $\sigma(e)(f(u)) \ne f(v)$. The concept of $S$-$k$-colourable is a common generalization of many colouring concepts. This paper studies the problem as to which subsets $S$ of $S_4$, every planar graph is $S$-$4$-colourable. We call such a subset $S$ of $S_4$ a good subset. The Four Colour Theorem is equivalent to saying that $S=\{id\}$ is good. It was proved by Jin, Wong and Zhu (arXiv:1811.08584) that a subset $S$ containing $id$ is good if and only if $S=\{id\}$. In this paper, we prove that, up to conjugation, every good subset of $S_4$ not containing $id$ is a subset of $\{(12),(34),(12)(34)\}$.

Sorting with Pattern-Avoiding Stacks: The 132-Machine

This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed by the above mentioned authors. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.

Zero-one Schubert polynomials

We prove that if $$\sigma \in S_m$$ σ ∈ S m is a pattern of $$w \in S_n$$ w ∈ S n , then we can express the Schubert polynomial $$\mathfrak {S}_w$$ S w as a monomial times $$\mathfrak {S}_\sigma $$ S σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $$\mathfrak {S}_w$$ S w being zero-one. In this case, the Schubert polynomial $$\mathfrak {S}_w$$ S w is equal to the integer point transform of a generalized permutahedron.