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


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