A298248 Triangle of double-Eulerian numbers DE(n,k) (n >= 0, 0 <= k <= max(0, 2*(n-1))) read by rows.

1, 1, 1, 0, 1, 1, 0, 4, 0, 1, 1, 0, 10, 2, 10, 0, 1, 1, 0, 20, 12, 54, 12, 20, 0, 1, 1, 0, 35, 42, 212, 140, 212, 42, 35, 0, 1, 1, 0, 56, 112, 675, 880, 1592, 880, 675, 112, 56, 0, 1, 1, 0, 84, 252, 1845, 3962, 9246, 9540, 9246, 3962, 1845, 252, 84, 0, 1

Offset

0,8

Comments

DE(n,k) = number of permutations with d descents and e descents of the inverse such that d+e = k.

References

Christian Stump, On bijections between 231-avoiding permutations and Dyck paths, MathSciNet:2734176

Links

Table of n, a(n) for n=0..64.

Dominique Foata and Guo-Niu Han, The q-series in Combinatorics; permutation statistics

FindStat - Combinatorial Statistic Finder, The sum of the number of descents and the number of recoils of a permutation

Example

The triangle DE(n, k) begins:
n\k 0    1     2     3      4      5      6     7     8    9   10
0:  1
1:  1
2:  1    0     1
3:  1    0     4     0      1
4:  1    0    10     2     10      0      1
5:  1    0    20    12     54     12     20     0     1
6:  1    0    35    42    212    140    212    42    35    0    1

Prog

(SageMath)
q = var("q")
[sum( q^(pi.number_of_descents()+pi.inverse().number_of_descents()) for pi in Permutations(n) ).coefficients(sparse=False) for n in [1 .. 6]]

Crossrefs

Row sums give A000142.

Cf. A000292, A008292, A180389.

Sequence in context: A059064 A321316 A185690 * A250204 A096459 A378833

Adjacent sequences: A298245 A298246 A298247 * A298249 A298250 A298251

Keyword

nonn,tabf

Author

Christian Stump, Jan 16 2018

Status

approved