## Information on Genocchi Numbers and Dumont Permutations

The Genocchi numbers, g1,g2,... can be defined in a number of different ways. There is a recurrence relation (with g1 = 1)

gn   =
 __ \ / i > 0
(-1)i ( n
2i
) gn-i.

The absolute value of the coefficient of the x2n/(2n)! term in the exponential generating function shown below is gn.

 2x ex+1
=     x   -   1
 x2 2!
+   1
 x4 4!
-   3
 x6 6!
+   17
 x8 8!
-   155
 x10 10!
+   · · ·

The first 15 Genocchi numbers are: 1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905, 51943281731, 2905151042481, 191329672483963, 14655626154768697, 1291885088448017715. This is sequence Anum=A001469"> A001469(M3041) in

Dominique Dumont showed that certain classes of permuations are counted by the Genocchi numbers. Dumont showed that the (n + 1)st Genocchi number is the number of permutations of [2n] with the following properties:

1. each even integer must be followed by a smaller integer. (This rule disallows the sequence from ending with an even integer)
2. each odd integer is either followed by a larger integer or is final in the sequence.
We call these Dumont permutations of the first kind. Dumont defined another type of permutation of [2n] and showed that they are also counted by the Genocchi numbers. These permutations p[1..2n] have the following properties and are called the Dumont permutations of the second kind.
1. For an even position i, the value of p[i] is less than i.
2. For an odd position i, the value of p[i] is at least i.
The table below shows all 17 permutations of both types for n=3. It is these permutations that are output by COS.

First Kind         Second Kind
2 1 4 3 6 5 4 1 5 2 6 3
2 1 5 6 4 3 3 1 5 2 6 4
2 1 6 4 3 5 5 1 4 2 6 3
3 4 2 1 6 5 3 1 4 2 6 5
3 5 6 4 2 1 5 1 3 2 6 4
3 6 4 2 1 5 4 1 3 2 6 5
4 2 1 3 6 5 4 1 5 3 6 2
4 2 1 5 6 3 2 1 5 3 6 4
4 2 1 6 3 5 5 1 4 3 6 2
4 3 5 6 2 1 2 1 4 3 6 5
4 3 6 2 1 5 4 1 6 2 5 3
5 6 2 1 4 3 3 1 6 2 5 4
5 6 3 4 2 1 6 1 4 2 5 3
5 6 4 2 1 3 6 1 3 2 5 4
6 2 1 4 3 5 4 1 6 3 5 2
6 3 4 2 1 5 2 1 6 3 5 4
6 4 2 1 3 5 6 1 4 3 5 2

### References

• A short biography of the Italian mathematician Angelo Genocchi (1870-1889).
• D. Dumont, "Intérpretation combinatoire des nombres de Genocchi," Duke J. Math., 41 (1974) 305-318.
• G. Kreweras, "An additive generation for the Genocchi numbers and two of its enumerative meanings", Bulletin of the ICA, 20 (1997) 99-103.
• A. Randrianarivony and J. Zeng, "Some equidistributed statistics on Genocchi permutations," Electronic J. Combinatorics, 3 (1996) R22.

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