Combinatorial properties of poly-Bernoulli relatives. (English) Zbl 1412.11034
Summary: In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show connections between the different areas where poly-Bernoulli numbers and their relatives appear and give examples of how the combinatorial methods can be used for deriving formulas between these integer arrays.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 05A19 | Combinatorial identities, bijective combinatorics |
Keywords:
poly-Bernoulli relativesOnline Encyclopedia of Integer Sequences:
3*Sum_{k>=0} k^n/binomial(2*k, k) = Pi*sqrt(3)*q(n) + a(n) for some rational sequence (q(n)).Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).
Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,k+n} having excedance set {1,2,...,k} (the empty set for k=0), 0 <= k <= n-1.
Number of permutations of {1,2,...,n} having excedance set {1,2,...,k} for some k=0...n-1 (for k=0 we have the empty set). The excedance set of a permutation p in S_n is the set of indices i such that p(i)>i.
Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.
Array read by antidiagonals: The number of parades with n girls and k boys that begin with a girl and end with a boy.
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