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Triangle of double-Eulerian numbers DE(n,k) (n >= 0, 0 <= k <= max(0, 2*(n-1))) read by rows.
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%I #45 Feb 26 2024 19:38:23

%S 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,

%T 212,140,212,42,35,0,1,1,0,56,112,675,880,1592,880,675,112,56,0,1,1,0,

%U 84,252,1845,3962,9246,9540,9246,3962,1845,252,84,0,1

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

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

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

%H Dominique Foata and Guo-Niu Han, <a href="http://irma.math.unistra.fr/~guoniu/papers/index.html">The q-series in Combinatorics; permutation statistics</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000824/">The sum of the number of descents and the number of recoils of a permutation</a>

%e The triangle DE(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10

%e 0: 1

%e 1: 1

%e 2: 1 0 1

%e 3: 1 0 4 0 1

%e 4: 1 0 10 2 10 0 1

%e 5: 1 0 20 12 54 12 20 0 1

%e 6: 1 0 35 42 212 140 212 42 35 0 1

%o (SageMath)

%o q = var("q")

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

%Y Row sums give A000142.

%Y Cf. A000292, A008292, A180389.

%K nonn,tabf

%O 0,8

%A _Christian Stump_, Jan 16 2018