scholarly journals The sorting index and inversion number on order ideals of permutation groups

2016 ◽  
Vol 339 (10) ◽  
pp. 2490-2499
Author(s):  
Neil J.Y. Fan ◽  
Liao He ◽  
Teresa X.S. Li ◽  
Alina F.Y. Zhao
Keyword(s):  
Permutation Groups ◽  
Inversion Number ◽  
Order Ideals ◽  
And Inversion

scholarly journals A Recurrence Relation for the "inv" Analogue of $q$-Eulerian Polynomials

10.37236/471 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Chak-On Chow
Keyword(s):  
Symmetric Group ◽  
Recurrence Relation ◽  
Combinatorial Proof ◽  
Algebraic Proof ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Inversion Number ◽  
And Inversion

We study in the present work a recurrence relation, which has long been overlooked, for the $q$-Eulerian polynomial $A_n^{{\rm des},{\rm inv}}(t,q) =\sum_{\sigma\in\mathfrak{S}_n} t^{{\rm des}(\sigma)}q^{{\rm inv}(\sigma)}$, where ${\rm des}(\sigma)$ and ${\rm inv}(\sigma)$ denote, respectively, the descent number and inversion number of $\sigma$ in the symmetric group $\mathfrak{S}_n$ of degree $n$. We give an algebraic proof and a combinatorial proof of the recurrence relation.

scholarly journals A Bijective Proof of a Major Index Theorem of Garsia and Gessel

10.37236/336 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mordechai Novick
Keyword(s):  
Generating Function ◽  
Index Theorem ◽  
Bijective Proof ◽  
Major Index ◽  
Inversion Number ◽  
And Inversion ◽  
Special Case

In this paper we provide a bijective proof of a theorem of Garsia and Gessel describing the generating function of the major index over the set of all permutations of $[n]=\{1,...,n\}$ which are shuffles of given disjoint ordered sequences $\pi_1,...,\pi_k$ whose union is $[n]$. The proof is based on a result (an "insertion lemma") of Haglund, Loehr, and Remmel which describes the change in major index resulting from the insertion of a given new element in any place in a given permutation. Using this lemma we prove the theorem by establishing a bijection between shuffles of ordered sequences and a certain set of partitions. A special case of Garsia and Gessel's theorem provides a proof of the equidistribution of major index and inversion number over inverse descent classes, a result first proved bijectively by Foata and Schutzenberger in 1978. We provide, based on the method of our first proof, another bijective proof of this result.

Major Index and Inversion Number of Permutations

1978 ◽  
Vol 83 (1) ◽  
pp. 143-159 ◽  
Author(s):  
Dominique Foata ◽  
Marcel-Paul Schützenberger
Keyword(s):  
Major Index ◽  
Inversion Number ◽  
And Inversion

The Fine Structure of the Sheath of Volvox Carteri f. Weismannia

1977 ◽  
Vol 35 ◽  
pp. 346-347
Author(s):  
Regina Birchem
Keyword(s):  
Developmental Stages ◽  
Ruthenium Red ◽  
Sulfated Polysaccharides ◽  
Volvox Carteri ◽  
High Voltage Electron Microscopy ◽  
Ultrastructural Studies ◽  
Voltage Electron ◽  
Sheath Formation ◽  
And Inversion ◽  
Sheath Material

Spheroids of the green colonial alga Volvox consist of biflagellate Chlamydomonad-like cells embedded in a transparent sheath. The sheath, important as a substance through which metabolic materials, light, and the sexual inducer must pass to and from the cells, has been shown to have an ordered structure (1,2). It is composed of both protein and carbohydrate (3); studies of V. rousseletii indicate an outside layer of sulfated polysaccharides (4).Ultrastructural studies of the sheath material in developmental stages of V. carteri f. weismannia were undertaken employing variations in the standard fixation procedure, ruthenium red, diaminobenzidine, and high voltage electron microscopy. Sheath formation begins after the completion of cell division and inversion of the daughter spheroids. Golgi, rough ER, and plasma membrane are actively involved in phases of sheath synthesis (Fig. 1). Six layers of ultrastructurally differentiated sheath material have been identified.

Exsolution and inversion in pigeonite from the Whin Sill, Northern England

1978 ◽  
Vol 36 (1) ◽  
pp. 474-475
Author(s):  
P.P.K. Smith
Keyword(s):  
High Temperature ◽  
Low Temperature ◽  
Homogeneous Nucleation ◽  
Silicate Mineral ◽  
Antiphase Boundaries ◽  
Two Phase ◽  
Primitive Form ◽  
The Core ◽  
Nucleation Mechanisms ◽  
And Inversion

Grains of pigeonite, a calcium-poor silicate mineral of the pyroxene group, from the Whin Sill dolerite have been ion-thinned and examined by TEM. The pigeonite is strongly zoned chemically from the composition Wo8En64FS28 in the core to Wo13En34FS53 at the rim. Two phase transformations have occurred during the cooling of this pigeonite:- exsolution of augite, a more calcic pyroxene, and inversion of the pigeonite from the high- temperature C face-centred form to the low-temperature primitive form, with the formation of antiphase boundaries (APB's). Different sequences of these exsolution and inversion reactions, together with different nucleation mechanisms of the augite, have created three distinct microstructures depending on the position in the grain.In the core of the grains small platelets of augite about 0.02μm thick have farmed parallel to the (001) plane (Fig. 1). These are thought to have exsolved by homogeneous nucleation. Subsequently the inversion of the pigeonite has led to the creation of APB's.

Compact cellular algebras and permutation groups

1999 ◽  
Vol 197-198 (1-3) ◽  
pp. 247-267 ◽  
Author(s):  
S Evdokimov
Keyword(s):  
Permutation Groups ◽  
Cellular Algebras

Analysis of the Bias Dependent Spectral Response of a-SiC:H p-i-n Photodiode

2002 ◽  
Vol 715 ◽  
Author(s):  
P. Louro ◽  
A. Fantoni ◽  
Yu. Vygranenko ◽  
M. Fernandes ◽  
M. Vieira
Keyword(s):  
Bias Voltage ◽  
Voltage Dependence ◽  
Spectral Response ◽  
Surface Recombination ◽  
Electrical Field ◽  
Field Reversal ◽  
Current Voltage ◽  
Voltage Dependent ◽  
And Inversion ◽  
Device Operation

AbstractThe bias voltage dependent spectral response (with and without steady state bias light) and the current voltage dependence has been simulated and compared to experimentally obtained values. Results show that in the heterostructures the bias voltage influences differently the field and the diffusion part of the photocurrent. The interchange between primary and secondary photocurrent (i. e. between generator and load device operation) is explained by the interaction of the field and the diffusion components of the photocurrent. A field reversal that depends on the light bias conditions (wavelength and intensity) explains the photocurrent reversal. The field reversal leads to the collapse of the diode regime (primary photocurrent) launches surface recombination at the p-i and i-n interfaces which is responsible for a double-injection regime (secondary photocurrent). Considerations about conduction band offsets, electrical field profiles and inversion layers will be taken into account to explain the optical and voltage bias dependence of the spectral response.

Selective Tracking Using Linear Trackability Analysis and Inversion-based Tracking Control

2020 ◽  
Author(s):  
Sujay D. Kadam ◽  
Aishwarya Rao ◽  
Biswajit Prusty ◽  
Harish J. Palanthandalam-Madapusi
Keyword(s):  
Tracking Control ◽  
And Inversion
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