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A298248
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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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0
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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
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OFFSET
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0,8
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COMMENTS
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DE(n,k) = number of permutations with d descents and e descents of the inverse such that d+e = k.
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REFERENCES
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Christian Stump, On bijections between 231-avoiding permutations and Dyck paths, MathSciNet:2734176
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LINKS
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EXAMPLE
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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
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PROG
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(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]]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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