scholarly journals A Generalization of Simion-Schmidt's Bijection for Restricted Permutations

10.37236/1686 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Astrid Reifegerste
Keyword(s):  
Symmetric Group ◽  
Permutation Statistics ◽  
Restricted Permutations ◽  
One To One ◽  
Special Case

We consider the two permutation statistics which count the distinct pairs obtained from the final two terms of occurrences of patterns $\tau_1\cdots\tau_{m-2}m(m-1)$ and $\tau_1\cdots\tau_{m-2}(m-1)m$ in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As a special case we derive a one-to-one correspondence between permutations which avoid each of the patterns $\tau_1\cdots\tau_{m-2}m(m-1)\in{\cal S}_m$ and those which avoid each of the patterns $\tau_1\cdots\tau_{m-2}(m-1)m\in{\cal S}_m$. For $m=3$ this correspondence coincides with the bijection given by Simion and Schmidt in [Europ. J. Combin. 6 (1985), 383-406].

scholarly journals Classification of fused links

2016 ◽  
Vol 25 (14) ◽  
pp. 1650076 ◽  
Author(s):  
Timur Nasybullov
Keyword(s):  
Symmetric Group ◽  
Equivalence Classes ◽  
One To One ◽  
Text Component

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of [Formula: see text]-component fused links is in one-to-one correspondence with the set of elements of the abelization [Formula: see text] up to conjugation by elements from the symmetric group [Formula: see text].

scholarly journals Congruences induced by transitive representations of inverse semigroups

1987 ◽  
Vol 29 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Mario Petrich ◽  
Stuart Rankin
Keyword(s):  
General Theory ◽  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transitive Group ◽  
Inverse Semigroups ◽  
Group Representations ◽  
Symmetric Inverse Semigroup ◽  
The Right ◽  
Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

Isochronous anharmonic oscillations

1998 ◽  
Vol 76 (8) ◽  
pp. 645-657 ◽  
Author(s):  
Pirooz Mohazzabi
Keyword(s):  
General Equation ◽  
Vertical Plane ◽  
One Dimensional ◽  
One To One ◽  
Anharmonic Oscillations ◽  
Special Case

The problem of a particle oscillating without friction on a curve in a vertical plane (referred to as a vertical curve) is addressed. It is shown that there are infinitely many asymmetric concave vertical curves on which oscillations of a particle remain isochronous. The general equation of these curves is derived, and a one-to-one correspondence between these curves and one-dimensional potentials is established. The results are compared with the existing literature, and an interesting nontrivial special case is discussed. Some issues regarding interpretation of the results in the context of action and angle variables are also addressed. PACS No. 03.20

scholarly journals Multiplicities of Higher Lie Characters

2003 ◽  
Vol 75 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Associative Algebra ◽  
Free Associative Algebra ◽  
Combinatorial Description ◽  
Irreducible Components ◽  
Witt Basis ◽  
Special Case ◽  
Regular Character

AbstractThe higher Lie characters of the symmetric group Sn arise from the Poincaré-Birkhoff-Witt basis of the free associative algebra. They are indexed by the partitions of n and sum up to the regular character of Sn. A combinatorial description of the multiplicities of their irreducible components is given. As a special case the Kraśkiewicz-Weyman result on the multiplicities of the classical Lie character is obtained.

scholarly journals Generating Functions for Permutations which Contain a Given Descent Set

10.37236/299 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Symmetric Function ◽  
Schur Functions ◽  
Permutation Statistics ◽  
Finite Set ◽  
Set Of Permutations ◽  
Descent Set ◽  
Positive Integers

A large number of generating functions for permutation statistics can be obtained by applying homomorphisms to simple symmetric function identities. In particular, a large number of generating functions involving the number of descents of a permutation $\sigma$, $des(\sigma)$, arise in this way. For any given finite set $S$ of positive integers, we develop a method to produce similar generating functions for the set of permutations of the symmetric group $S_n$ whose descent set contains $S$. Our method will be to apply certain homomorphisms to symmetric function identities involving ribbon Schur functions.

scholarly journals The semigroup of one-to-one transformations with finite defects

1989 ◽  
Vol 31 (2) ◽  
pp. 243-249 ◽  
Author(s):  
Inessa Levi ◽  
Boris M. Schein
Keyword(s):  
Symmetric Group ◽  
Disjoint Union ◽  
One To One ◽  
Infinite Set ◽  
Number Of Authors

Let be the semigroup of all total one-to-one transformations of an infinite set X. For an ƒ ∈ let the defect of ƒ def ƒ, be the cardinality of X – R(ƒ), where R(ƒ) = ƒ(X) is the range of ƒ. Then is a disjoint union of the symmetric group x on X, the semigroup S of all transformations in with finite non-zero defects and the semigroup Ā of all transformations in S with infinite defects, such that S U Ā and Ā are ideals of . The properties of x and Ā have been investigated by a number of authors (for the latter it was done via Baer-Levi semigroups, see [2], [3], [5], [6], [7], [8], [9], [10] and note that Ā decomposes into a union of Baer–Levi semigroups). Our aim here is to study the semigroup S. It is not difficult to see that S is left cancellative (we compose functions ƒ, g in S as ƒg(x) = ƒ(g(x)), for x ∈ X) and idempotent-free. All automorphisms of S are inner [4], that is of the form ƒ → hƒhfh-1 ƒ ∈ S, h ∈ x.

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

2000 ◽  
Vol 43 (4) ◽  
pp. 448-458
Author(s):  
Chi-Kwong Li ◽  
Alexandru Zaharia
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Numerical Range ◽  
Complex Matrix ◽  
Normal Matrices ◽  
Principal Character ◽  
Straight Line ◽  
Generalized Matrix ◽  
Generalized Numerical Range ◽  
Special Case

AbstractSuppose m and n are integers such that 1 ≤ m ≤ n. For a subgroup H of the symmetric group Sm of degree m, consider the generalized matrix function on m × m matrices B = (bij) defined by and the generalized numerical range of an n × n complex matrix A associated with dH defined byIt is known that WH(A) is convex if m = 1 or if m = n = 2. We show that there exist normal matrices A for which WH(A) is not convex if 3 ≤ m ≤ n. Moreover, for m = 2 < n, we prove that a normal matrix A with eigenvalues lying on a straight line has convex WH(A) if and only if νA is Hermitian for some nonzero ν ∈ ℂ. These results extend those of Hu, Hurley and Tam, who studied the special case when 2 ≤ m ≤ 3 ≤ n and H = Sm.

Multiple-Answer Items and Chance Success

1965 ◽  
Vol 16 (3_suppl) ◽  
pp. 1011-1012
Author(s):  
Lewis R. Aiken
Keyword(s):  
General Formula ◽  
Correct Answer ◽  
Test Item ◽  
Hypergeometric Distribution ◽  
Objective Test ◽  
Occupancy Problem ◽  
One To One ◽  
Special Case ◽  
Chance Success

The use of a general formula, the solution to a special case of the classical occupancy problem, for estimating the probability of chance success on any one-to-one objective test item is reviewed. It is noted that it may on occasion be more appropriate to write items with more than one correct answer and that the chance success formula for this situation is the hypergeometric distribution.

scholarly journals Symmetric spectral synthesis

2020 ◽  
Author(s):  
Eszter Gselmann ◽  
László Székelyhidi
Keyword(s):  
Symmetric Group ◽  
Spectral Synthesis ◽  
Higher Dimensions ◽  
Acta Math ◽  
Continuous Functions ◽  
Gelfand Pairs ◽  
Space Of Continuous Functions ◽  
Translation Invariant ◽  
Special Case ◽  
Straightforward Extension

AbstractAccording to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d.

scholarly journals Cycles and sorting index for matchings and restricted permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović
Keyword(s):  
Permutation Statistics ◽  
Minimal Elements ◽  
Restricted Permutations ◽  
Dyck Paths ◽  
Set Of Permutations ◽  
Ferrers Board ◽  
International Audience ◽  
The Right

International audience We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.

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