Abstract
It is proved that the number of n-element permutationally-ordered sets with the maximal anti-chain of length k is not greater than \(\min \left\{ {\tfrac{{k^{2n} }} {{(k!)^2 }},\tfrac{{(n - k + 1)^{2n} }} {{((n - k)!)^2 }}} \right\}\). It is also proved that the number of permutations £ k (n) of the numbers {1,..., n} with the maximal decreasing subsequence of length at most k satisfies the inequality \(\tfrac{{k^{2n} }} {{((k - 1)!)^2 }}\). A review of papers focused on bijections and relations between pairs of linear orders, pairs of Young diagrams, two-dimensional integer arrays, and integer matrices is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Schensted, “Longest Increasing and Decreasing Subsequences,” Can. J. Math. 13, 179 (1961).
V. N. Latyshev, “On Regev’s Theorem on Identities in a Tensor Product of -P/-algebras,” Uspekhi Matem. Nauk 27 (4), 213 (1972).
A. Ya. Belov and M. I. Kharitonov, “Shirshov?s Height Estimates and Estimates on Numbers of Periodic Parts of Small Periods,” Fundament. Prikl. Matem. 17 (5), 21, (2012) [“Subexponential Estimates in the Height Theorem and Estimates on Numbers of Periodic Parts of Small Periods,” J. Math. Sci. 193 (4), 493, (2013)].
G. R. Chelnokov, “On a Lower Estimate of Numbers of ( k + 1)-divisible Permutations,” Modeling and Analysis of Information Systems (MAIS) 14 (4), 53 (2007).
D. E. Knuth, “Permutations, Matrices, and Generalized Young Tableaux,” Pacif. J. Math. 34 (3), 709 (1970).
I. M. Gessel, “Symmetric Functions and P-Recursiveness,” J. Combin. Theory, Ser. A 53, 257, (1990).
W. Specht, “Gesetze in Ringen I,” Math. Z. 52, 557, (1950).
V. N. Latyshev, “Non-Matrix Varieties of Associative Algebras,” Doctoral Dissertation in Mathematics and Physics (Moscow State Univ., Moscow, 1977).
A. R. Kemer, “Finite Basis Property of Identities of Associative Algebras,” Algebra i Logika 26 (5), 597 (1987) [Algebra and Logic 26 (5), 362 (1987)]
A. Ya. Belov, “On Non-Spechtian Varieties,” Fundament. Prikl. Matem. 5 (1), 47 (1999).
A. V. Grishin, “Examples of T-Spaces and T-Ideals over a Field of Characteristic 2 without the Finite Basis Property,” Fundament. Prikl. Matem. 5 (1), 101 (1999).
V. V. Shchigoloev, “Examples of Infinitely Based T-ideals,” Fundament. Prikl. Matem. 5 (1), 307 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M.I. Kharitonov, 2015, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2015, Vol. 70, No. 3, pp. 24–28.
About this article
Cite this article
Kharitonov, M.I. The estimate of the number of permutationally-ordered sets. Moscow Univ. Math. Bull. 70, 125–129 (2015). https://doi.org/10.3103/S0027132215030055
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132215030055