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The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices.
- Source :
-
Discrete Mathematics . Nov2019, Vol. 342 Issue 11, p2966-2983. 18p. - Publication Year :
- 2019
-
Abstract
- In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials F m , n , 1 (x) and F m , n , 2 (x). Then, he defined A m (n , k) and B m (n , k) to be the polynomials satisfying F m , n , 1 (x) = ∑ k = 0 n A m (n , k) x n − k (x + 1) k and F m , n , 1 (x) = ∑ k = 0 n B m (n , k) x n − k (x + 1) k. In this paper, we give a combinatorial interpretation of the coefficients of A m + 1 (n , k) and prove a symmetry of the coefficients, i.e., [ m s ] A m + 1 (n , k) = [ m n − s ] A m + 1 (n , n − k). We give a combinatorial interpretation of B m + 1 (n , k) and prove that B m + 1 (n , n − 1) is a polynomial in m with non-negative integer coefficients. We also prove that if n ≥ 6 then all coefficients of B m + 1 (n , n − 2) except the coefficient of m n − 1 are non-negative integers. For all n , the coefficient of m n − 1 in B m + 1 (n , n − 2) is − (n − 1) , and when n ≤ 5 some other coefficients of B m + 1 (n , n − 2) are also negative. [ABSTRACT FROM AUTHOR]
- Subjects :
- *POLYNOMIALS
*COMBINATORICS
*EULER'S numbers
*INTEGERS
Subjects
Details
- Language :
- English
- ISSN :
- 0012365X
- Volume :
- 342
- Issue :
- 11
- Database :
- Academic Search Index
- Journal :
- Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 138252596
- Full Text :
- https://doi.org/10.1016/j.disc.2019.05.026