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Regular version of the site

Article

Symmetric Dellac Configurations

Journal of Integer Sequences. 2020. Vol. 23. No. 20.4.6. P. 1-32.
Bigeni A., Feigin E.

We define symmetric Dellac configurations as the Dellac configurations that are symmetrical with respect to their centers. The even-length symmetric Dellac configurations coincide with the Fang-Fourier symplectic Dellac configurations. Symmetric Dellac configurations generate the Poincaré polynomials of (odd or even) symplectic or orthogonal versions of degenerate flag varieties. We give several combinatorial interpretations of the Randrianarivony-Zeng polynomial extension of median Euler numbers in terms of objects that we call extended Dellac configurations. We show that the extended Dellac configurations generate symmetric Dellac configurations. As a consequence, the cardinalities of odd and even symmetric Dellac configurations are respectively given by two sequences (1, 1, 3, 21, 267, ...) and (1, 2, 10, 98, 1594, ...), defined as specializations of polynomial extensions of median Euler numbers.