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 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS T(n,k) is the number of derangements of [n] with k descents. LINKS Shishuo Fu, Z. Lin, J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184 [math.CO], 2015-2019. FORMULA The g.f. F(x,y) = Sum_{n>=2,1<=k<=n-1}T(n,k)x^n/n!y^k satisfies the partial differential equation (1-xy) D_{x}F + (y^2-y) 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 == n-1 := 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 A348874 A300328 Adjacent sequences:  A219833 A219834 A219835 * A219837 A219838 A219839 KEYWORD nonn,tabl AUTHOR David Callan, Nov 29 2012 STATUS approved

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Last modified August 13 22:39 EDT 2022. Contains 356109 sequences. (Running on oeis4.)