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

Author

Christian Stump, Jan 16 2018

Keywords

• nonn
• tabf

Comments

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

Examples

			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
		

References

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

Links

• 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

Crossrefs

Row sums give A000142.

Cf. A000292, A008292, A180389.

Programs

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]]