scholarly journals Polyurethane Toggles

10.37236/9097 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Colin Defant
Keyword(s):  
Simple Proof ◽  
Group Structure ◽  
Fundamental Properties ◽  
Interesting Connection ◽  
Stack Sorting ◽  
Wilf Equivalence

We consider the involutions known as toggles, which have been used to give simplified proofs of the fundamental properties of the promotion and evacuation maps. We transfer these involutions so that they generate a group $\mathscr P_n$ that acts on the set $S_n$ of permutations of $\{1,\ldots,n\}$. After characterizing its orbits in terms of permutation skeletons, we apply the action in order to understand West's stack-sorting map. We obtain a very simple proof of a result that clarifies and extensively generalizes a theorem of Bouvel and Guibert and also generalizes a theorem of Bousquet-M\'elou. We also settle a conjecture of Bouvel and Guibert. We prove a result related to the recently-introduced notion of postorder Wilf equivalence. Finally, we investigate an interesting connection among the action of $\mathscr P_n$ on $S_n$, the group structure of $S_n$, and the stack-sorting map.

The Homotopy Set of the Axes of Pairings

1990 ◽  
Vol 42 (5) ◽  
pp. 856-868 ◽  
Author(s):  
Nobuyuki Oda
Keyword(s):  
Group Structure ◽  
Homotopy Classes ◽  
Fundamental Properties ◽  
Cyclic Map ◽  
Cyclic Maps ◽  
Image Position

Varadarajan [13] named a map f: A → X a cyclic map when there exists a map F: X × A → X such that for the folding map ∇X: X ∨ X → X. He defined the generalized Gottlieb set G(A, X) of the homotopy classes of the cyclic maps F: A → X and studied the fundamental properties of G(A, X) If A is a co-Hopf space, then the Varadarajan set G(A, X) has a group structure [13]. The group G(A,X) is a generalization of G(X) and Gn(X) of Gottlieb [2,3]. Some authors studied the properties of the Varadarajan set, its dual and related topics [4, 5, 6, 7,12,15,16,17].

Operational Systems and Checker Games

1967 ◽  
Vol 60 (8) ◽  
pp. 832-836
Author(s):  
Hans-Georg Steiner
Keyword(s):  
Group Theory ◽  
Secondary School ◽  
Group Structure ◽  
School Level ◽  
Fundamental Properties ◽  
Elementary Group ◽  
Secondary School Level ◽  
Operational Systems

In the teaching of graphs and elementary group theory at the secondary school level it is important to use intuitive and operative material in which group structure is involved. Working with materials can help to deepen the understanding or even lead the student to a first observation of the fundamental properties of a group.

On q-Analogues of Sumudu Transform

2013 ◽  
Vol 21 (1) ◽  
pp. 239-259 ◽  
Author(s):  
Durmuş Albayrak ◽  
Sunil Dutt Purohit ◽  
Faruk Uçar
Keyword(s):  
Infinite Series ◽  
Laplace Transforms ◽  
Convolution Theorem ◽  
Convergence Conditions ◽  
Fundamental Properties ◽  
Interesting Connection ◽  
Sumudu Transform ◽  
Sumudu Transforms

Abstract The present paper introduces q-analogues of the Sumudu transform and derives some distinct properties, for example its convergence conditions and certain interesting connection theorems involving q-Laplace transforms. Furthermore, certain fundamental properties of q-Sumudu transforms like, linearity, shifting theorems, differentiation and integration etc. have also been investigated. An attempt has also been made to obtain the convolution theorem for the q-Sumudu transform of a function which can be expressed as a convergent infinite series.

scholarly journals Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

2021 ◽  
Vol vol. 22 no. 2, Permutation... (Combinatorics) ◽  
Author(s):  
Colin Defant
Keyword(s):  
Explicit Formula ◽  
Catalan Numbers ◽  
Stack Sorting ◽  
Set Of Permutations ◽  
Order Ideals ◽  
Wilf Equivalence ◽  
Young’S Lattice

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

Some effects of unequal distribution of information in a wheel group structure.

1954 ◽  
Vol 49 (4, Pt.1) ◽  
pp. 554-556 ◽  
Author(s):  
J. C. Gilchrist ◽  
Marvin E. Shaw ◽  
L. C. Walker
Keyword(s):  
Group Structure ◽  
Unequal Distribution
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