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Boolean analogues of the Pascal triangle with maximal possible number of ones

  • Fedor M. Malyshev EMAIL logo
Published/Copyright: October 13, 2021
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 31 Issue 5

Abstract

The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array Ts formed by s(s+1)2 elements of the field GF(2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ s(s+1)3 ⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.

Keywords: Pascal triangles; (0, 1)-matrices; extremal combinatorial configurations

Originally published in Diskretnaya Matematika (2020) 32,№1, 51–59 (in Russian).

References

[1] Malyshev F. M., “Bases of integers under the multiplace shift operations”, Matematicheskie voprosy kriptografii, 2:1 (2011), 29-74 (in Russian).10.4213/mvk25Search in Google Scholar

[2] Malyshev F. M., “Bases of recurrent sequences”, Chebyshevskiy sbornik, 16:2 (2015), 155-185 (in Russian).Search in Google Scholar

[3] Wolfram S., Cellular Automaton Supercomputing. In High-Speed Computing, University of Illinois Press, 1988.Search in Google Scholar

[4] Malyshev F. M., Kutyreva E. V., “On the distribution of the number of ones in a Boolean Pascal’s triangle”, Discrete Math. Appl, 16:3 (2006), 271-279.10.1515/156939206777970435Search in Google Scholar

[5] Malyshev F.M., “Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle”, Discrete Math. Appl., 27:3 (2017), 149-176.10.1515/dma-2017-0019Search in Google Scholar
Received: 2019-12-19
Published Online: 2021-10-13
Published in Print: 2021-10-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Diagnostic tests under shifts with fixed filling tuple
  3. On the average-case complexity of Boolean functions under binomial distribution on their domains
  4. Boolean analogues of the Pascal triangle with maximal possible number of ones
  5. The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups
  6. On the complexity of monotone circuits for threshold symmetric Boolean functions
  7. Properties of multitype subcritical branching processes in random environment
https://doi.org/10.1515/dma-2021-0029
Keywords for this article
Pascal triangles; (0, 1)-matrices; extremal combinatorial configurations

Articles in the same Issue

  1. Frontmatter
  2. Diagnostic tests under shifts with fixed filling tuple
  3. On the average-case complexity of Boolean functions under binomial distribution on their domains
  4. Boolean analogues of the Pascal triangle with maximal possible number of ones
  5. The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups
  6. On the complexity of monotone circuits for threshold symmetric Boolean functions
  7. Properties of multitype subcritical branching processes in random environment
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