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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
|
=
|
|
(-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.
|
|
=
|
x -
|
1
|
|
+ 1
|
|
- 3
|
|
+ 17
|
|
- 155
|
|
+ · · ·
|
The first 15 Genocchi numbers are: 1, 1, 3, 17, 155, 2073, 38227, 929569,
28820619, 1109652905, 51943281731, 2905151042481, 191329672483963,
14655626154768697, 1291885088448017715.
This is sequence
A001469(M3041) in
Neil J. Sloane's
database
of integer sequences.
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:
- each even integer must be followed by a smaller integer. (This rule
disallows the sequence from ending with an even integer)
- 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.
-
For an even position i, the value of p[i] is less than
i.
-
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.
Programs available: 
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It was last updated Wednesday, 10-May-2006 10:32:14 PDT.
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