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Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2015, Number 3, Pages 24–28 (Mi vmumm235)  

Mathematics

The estimate of the number of permutationally-ordered sets

M. I. Kharitonov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: It is proved that the number of $n$-element permutationally-ordered sets with the maximal antichain of length not exceeding $k$ is not greater than $\min\{{k^{2n}\over (k!)^2}, {(n-k+1)^{2n}\over ((n-k)!)^2}\}$. It is also proved that the number of permutations $\xi_k(n)$ of the numbers $\{1,…,n\}$ with the maximal decreasing subsequence of length not exceeding $k$ satisfies the inequality ${k^{2n}\over ((k-1)!)^2}.$ A review of papers focused on bijections and relations between pairs of linear orders, pairs of Young diagrams, two-dimensional arrays of positive integers, and matrices with integer elements is presented.

Key words: combinatorics on words, $k$-divisibility, Dilworth theorem, multilinear words, multilinear identities, Young diagrams.

Funding Agency Grant Number
Simons Foundation
Dynasty Foundation
Russian Foundation for Basic Research 14-01-00548


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English version:
Moscow University Mathematics Bulletin, 2015, 70:3, 125–129

Bibliographic databases:

UDC: 512.562+519.1
Received: 09.12.2013

Citation: M. I. Kharitonov, “The estimate of the number of permutationally-ordered sets”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2015, no. 3, 24–28; Moscow University Mathematics Bulletin, 70:3 (2015), 125–129

Citation in format AMSBIB
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\issue 3
\pages 24--28
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