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Asymptotic Behavior of Perturbations of Symmetric Functions

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Abstract

In this paper we consider perturbations of symmetric Boolean functions \({{\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}}}\) in n-variable and degree k s . We compute the asymptotic behavior of Boolean functions of the type

$${\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}} +F(X_1, . . . , X_j)$$

for j fixed. In particular, we characterize all the Boolean functions of the type

$${\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}} +F(X_1, . . . , X_j)$$

that are asymptotic balanced. We also present an algorithm that computes the asymptotic behavior of a family of Boolean functions from one member of the family. Finally, as a byproduct of our results, we provide a relation between the parity of families of sums of binomial coefficients.

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Authors and Affiliations

  1. Department of Mathematics, University of Puerto Rico, San Juan, PR, 00931, USA

    Francis N. Castro & Luis A. Medina

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  1. Francis N. Castro
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  2. Luis A. Medina
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Correspondence to Luis A. Medina.

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Castro, F.N., Medina, L.A. Asymptotic Behavior of Perturbations of Symmetric Functions. Ann. Comb. 18, 397–417 (2014). https://doi.org/10.1007/s00026-014-0230-0

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  • DOI: https://doi.org/10.1007/s00026-014-0230-0

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