scholarly journals Counting Parabolic Double Cosets in Symmetric Groups

10.37236/9988 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Thomas Browning
Keyword(s):  
Asymptotic Formula ◽  
Polynomial Time ◽  
Coxeter Groups ◽  
Symmetric Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Double Cosets ◽  
New Formula

Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, and used it to compute $p_n$, the number of parabolic double cosets in $S_n$, for $n\leq13$. In this paper, we derive a new formula for $p_n$ and an efficient polynomial time algorithm for evaluating this formula. We use these results to compute $p_n$ for $n\leq5000$ and to prove an asymptotic formula for $p_n$ that was conjectured by Billey et al.

scholarly journals Non-commutative lattice problems

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Alexei Myasnikov ◽  
Andrey Nikolaev ◽  
Alexander Ushakov
Keyword(s):  
Computational Complexity ◽  
Large Class ◽  
Polynomial Time ◽  
Coxeter Groups ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Group Element ◽  
Surface Groups ◽  
Nontrivial Element

AbstractWe consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup element closest to a given group element, or finding a shortest nontrivial element of a subgroup in the case of nilpotent groups, and a large class of surface groups and Coxeter groups. We also provide polynomial time algorithm to compute geodesics in given generators of a subgroup of a free group.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

A polynomial-time algorithm for designing FIR filters with power-of-two coefficients

2002 ◽  
Vol 50 (8) ◽  
pp. 1935-1941 ◽  
Author(s):  
Dongning Li ◽  
Yong Ching Lim ◽  
Yong Lian ◽  
Jianjian Song
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fir Filters
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