

A219836


Triangular array counting derangements by number of descents.


2



1, 2, 0, 4, 4, 1, 8, 24, 12, 0, 16, 104, 120, 24, 1, 32, 392, 896, 480, 54, 0, 64, 1368, 5544, 5984, 1764, 108, 1, 128, 4552, 30384, 57640, 34520, 6048, 224, 0, 256, 14680, 153400, 470504, 495320, 180416, 19936, 448, 1
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OFFSET

2,2


COMMENTS

T(n,k) is the number of derangements of [n] with k descents.


LINKS

Table of n, a(n) for n=2..46.
Shishuo Fu, Z. Lin, J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184 [math.CO], 20152019.


FORMULA

The g.f. F(x,y) = Sum_{n>=2,1<=k<=n1}T(n,k)x^n/n!y^k satisfies the partial differential equation (1xy) D_{x}F + (y^2y) D_{y}F = F + 1  e^(xy). (Is there a closed form solution?)


EXAMPLE

Array begins
1
2, 0
4, 4, 1
8, 24, 12, 0
16, 104, 120, 24, 1
T(4,2) = 4 counts 2143, 3142, 3421, 4312.


MATHEMATICA

u[n_, 0] := 0; u[n_, k_] /; k == n1 := If [EvenQ[n], 1, 0]; u[n_, k_] /; 1 <= k <= n  2 := (n  k) u[n  1, k  1] + (k + 1) u[n  1, k]; Table[u[n, k], {n, 2, 10}, {k, n  1}]


CROSSREFS

Cf. A008292. (analogous for permutations)
Row sums give A000166. A046739 counts derangements of [n] by number of excedances.
Sequence in context: A167312 A114122 A319934 * A004174 A300328 A200291
Adjacent sequences: A219833 A219834 A219835 * A219837 A219838 A219839


KEYWORD

nonn,tabl


AUTHOR

David Callan, Nov 29 2012


STATUS

approved



