scholarly journals A bijective proof of Macdonald’s reduced word formula

2019 ◽  
Vol 2 (2) ◽  
pp. 217-248 ◽  
Author(s):  
Sara C. Billey ◽  
Alexander E. Holroyd ◽  
Benjamin J. Young
Keyword(s):  
Reduced Word ◽  
Bijective Proof

scholarly journals A bijective proof of Macdonald's reduced word formula

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Alexander Holroyd ◽  
Benjamin Young
Keyword(s):  
Reduced Word ◽  
Bijective Proof ◽  
International Audience ◽  
Principal Specialization

International audience We describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to Fomin and Stanley. Our bijective tools also allow us to address a problem posed by Fomin and Kirillov from 1997, using work of Wachs, Lenart and Serrano- Stump.

scholarly journals The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene

10.37236/1554 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Stefan Felsner
Keyword(s):  
Bruhat Order ◽  
Young Tableaux ◽  
Reduced Word ◽  
Reduced Decompositions ◽  
Bijective Proof ◽  
Identity Permutation ◽  
Weak Bruhat Order ◽  
Classical Correspondence ◽  
Standard Young Tableaux ◽  
Maximal Chains

Stanley conjectured that the number of maximal chains in the weak Bruhat order of $S_n$, or equivalently the number of reduced decompositions of the reverse of the identity permutation $ w_0 = n,n-1,n-2,\ldots,2,1$, equals the number of standard Young tableaux of staircase shape $s=\{n-1,n-2,\ldots,1\}$. Originating from this conjecture remarkable connections between standard Young tableaux and reduced words have been discovered. Stanley proved his conjecture algebraically, later Edelman and Greene found a bijective proof. We provide an extension of the Edelman and Greene bijection to a larger class of words. This extension is similar to the extension of the Robinson-Schensted correspondence to two line arrays. Our proof is inspired by Viennot's planarized proof of the Robinson-Schensted correspondence. As it is the case with the classical correspondence the planarized proofs have their own beauty and simplicity.

A bijective proof of a q-analogue of the sum of cubes using overpartitions

2017 ◽  
Vol 10 (3) ◽  
pp. 523-530
Author(s):  
Jacob Forster ◽  
Kristina Garrett ◽  
Luke Jacobsen ◽  
Adam Wood
Keyword(s):  
Bijective Proof

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals On the Distribution of Depths in Increasing Trees

10.37236/409 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Markus Kuba ◽  
Stephan Wagner
Keyword(s):  
Common Ancestor ◽  
Short Note ◽  
Bijective Proof ◽  
Lowest Common Ancestor ◽  
Recursive Trees

By a theorem of Dobrow and Smythe, the depth of the $k$th node in very simple families of increasing trees (which includes, among others, binary increasing trees, recursive trees and plane ordered recursive trees) follows the same distribution as the number of edges of the form $j-(j+1)$ with $j < k$. In this short note, we present a simple bijective proof of this fact, which also shows that the result actually holds within a wider class of increasing trees. We also discuss some related results that follow from the bijection as well as a possible generalization. Finally, we use another similar bijection to determine the distribution of the depth of the lowest common ancestor of two nodes.

scholarly journals The effect of speech situation on the occurrence of reduced word pronunciation variants

2015 ◽  
Vol 48 ◽  
pp. 60-75 ◽  
Author(s):  
Mirjam Ernestus ◽  
Iris Hanique ◽  
Erik Verboom
Keyword(s):  
Reduced Word ◽  
Speech Situation

scholarly journals Visual word form processing deficits driven by severity of reading impairments in children with developmental dyslexia

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
S. Brem ◽  
U. Maurer ◽  
M. Kronbichler ◽  
M. Schurz ◽  
F. Richlan ◽  
...  
Keyword(s):  
Reading Fluency ◽  
Word Processing ◽  
Visual Word ◽  
Common Finding ◽  
Word Form ◽  
Reduced Word ◽  
Form Processing ◽  
Visual Word Form Area ◽  
Intermediate Readers ◽  
Spatially Extended

Abstract The visual word form area (VWFA) in the left ventral occipito-temporal (vOT) cortex is key to fluent reading in children and adults. Diminished VWFA activation during print processing tasks is a common finding in subjects with severe reading problems. Here, we report fMRI data from a multicentre study with 140 children in primary school (7.9–12.2 years; 55 children with dyslexia, 73 typical readers, 12 intermediate readers). All performed a semantic task on visually presented words and a matched control task on symbol strings. With this large group of children, including the entire spectrum from severely impaired to highly fluent readers, we aimed to clarify the association of reading fluency and left vOT activation during visual word processing. The results of this study confirm reduced word-sensitive activation within the left vOT in children with dyslexia. Interestingly, the association of reading skills and left vOT activation was especially strong and spatially extended in children with dyslexia. Thus, deficits in basic visual word form processing increase with the severity of reading disability but seem only weakly associated with fluency within the typical reading range suggesting a linear dependence of reading scores with VFWA activation only in the poorest readers.

scholarly journals The Influence of Elision on the Original Syllabic Structure in English and Safwani Arabic: A Contextual Analysis

2016 ◽  
Vol 6 (2) ◽  
pp. 1
Author(s):  
Majid Abdulatif Ibrahim
Keyword(s):  
Vital Role ◽  
Contextual Analysis ◽  
Phonological Processes ◽  
Reduced Word ◽  
Syllabic Structure ◽  
Colloquial Speech

<p>Elision, as being a distinguishing mark among phonological processes, plays a vital role in patterning and mapping syllables of the language in a way that it can “distort” particular syllabic forms and templates. It represents deterioration, modification and to some extent radical changes in the syllabic structure of the original words even though it is usually a result of rapid colloquial speech. This study is a phonological work to detect the influence of sound deletion on syllabic templates and patterns of English and Safwani Arabic. It is an attempt to work out an analysis of the possible contexts where individual segments and syllables exhibit deletion. The analysis of all possible contexts where segments and syllables are lost in these varieties is illustrated by detailed tables within the paper. The tables are designed in a way that both original and reduced word or phrase forms are given, the context of sound elision is provided and then both original and resulting syllabic patterns are demonstrated.</p>

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