Interprétations combinatoires des nombres de Genocchi

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“…Moreover, since the Poincaré polynomials of Gr f (I ) and of Gr dim P− f (P) can be easily computed, we arrive at a formula for the Poincaré polynomial (and thus for the Euler characteristic) of X . Recall (see [Feigin 2010]) that the Euler characteristic of the variety F a n+1 is given by the normalized median Genocchi number h n+1 (see [Dellac 1900;Dumont 1974;Dumont and Randrianarivony 1994;Dumont and Zeng 1994;Viennot 1982]). Using Theorem 1.2 we obtain an explicit formula for h n+1 in terms of binomial coefficients.…”

Sur les Permutations Comptées par les Nombres de Genocchi de 1-ière et 2-ième Espèce

“…Les formules (1) et (2) , prises ensemble , constituent pre ´ cise ´ ment la re ` gle de formation du tableau T1 ; ce qui ache ` ve d'e ´ tablir la Proposition 1 .…”

“…Dans la Section 2 nous construirons par un algorithme additif un tableau triangulaire T1 , dans lequel les 2 n Ϫ 1 termes de la n -ie ` me ligne seront de ´ signe ´ s par g (2) g (3) и и и g (2 n ) et les 2 n ϩ 1 termes de la ( n ϩ 1)-ie ` me ligne par h (2) h (3) h (4) и и и h (2 n ϩ 1) h (2 n ϩ 2) . La re ` gle de formation est la suivante :…”

Further divisibility properties of the 𝑞-tangent numbers

“…The ^-tangent numbers are polynomials that may be defined by where G2n+2 is an odd integer called the Genocchi number (see e.g. [3]), Schützen-berger [6] raised the problem of finding a polynomial of the form II (1 + q')**0 I>1 that divides T2n+1(q). Along these lines Andrews and Gessel [2] proved that T2n+l(q) is divisible by…”