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On an extension of Euler numbers and records of alternating permutations. (Sur une extension des nombres d’Euler et les records des permutations alternantes.) (French. English summary) Zbl 0978.05502

Summary: We study the sequence of polynomials \(C_n(x,y)\) defined through the recurrence \(C_0(x,y)=1\), \(C_n(x,y)=x(y+1)C_{n-1}(x+2,y+2)-xyC_{n-1}(x,y)\), which turns out to be an extension of Euler numbers. We give a combinatorial interpretation of these numbers in terms of down-up permutations with respect to the numbers of even and odd upper records, and a continued fraction expansion for their ordinary generating function.
The paper has been finally published under the same title in [J. Comb. Theory, Ser. A 68, No. 1, 86-99 (1994; Zbl 0809.05002)].

MSC:

05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
11J70 Continued fractions and generalizations

Citations:

Zbl 0809.05002
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