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A186363 Triangle read by rows: T(n,k) is the number of cycle-up-down permutations of {1,2,...,n} having k fixed points (0 <= k <= n). A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1) < b(2) > b(3) < ... .

%I A186363 #11 Nov 03 2019 19:41:32
%S A186363 1,0,1,1,0,1,1,3,0,1,5,4,6,0,1,15,25,10,10,0,1,71,90,75,20,15,0,1,341,
%T A186363 497,315,175,35,21,0,1,1945,2728,1988,840,350,56,28,0,1,12135,17505,
%U A186363 12276,5964,1890,630,84,36,0,1,84091,121350,87525,40920,14910,3780,1050,120,45,0,1
%N A186363 Triangle read by rows: T(n,k) is the number of cycle-up-down permutations of {1,2,...,n} having k fixed points (0 <= k <= n). A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1) < b(2) > b(3) < ... .
%C A186363 Sum of entries in row n is A000111(n+1) (the Euler or up-down numbers).
%C A186363 T(n,0) = A186364(n).
%C A186363 T(n,k) = T(n-k,0)*binomial(n,k).
%C A186363 Sum_{k=0..n} k*T(n,k) = A186365(n).
%H A186363 E. Deutsch and S. Elizalde, <a href="http://arxiv.org/abs/0909.5199v1"> Cycle up-down permutations</a>, arXiv:0909.5199v1 [math.CO].
%F A186363 E.g.f. = exp((x-1)z)/(1-sin z).
%F A186363 The trivariate e.g.f. H(t,s,z) of the cycle-up-down permutations of {1,2,...,n} with respect to size (marked by z), number of cycles (marked by t), and number of fixed points (marked by x) is given by H(t,x,z) = exp((x-1)*t*z)/(1-sin(z))^t.
%e A186363 T(3,1)=3 because we have (1)(23), (12)(3), and (13)(2).
%e A186363 T(4,2)=6 because we have (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4), and (14)(2)(3).
%e A186363 Triangle starts:
%e A186363    1;
%e A186363    0,  1;
%e A186363    1,  0,  1;
%e A186363    1,  3,  0,  1;
%e A186363    5,  4,  6,  0,  1;
%e A186363   15, 25, 10, 10,  0,  1;
%p A186363 G := exp((x-1)*z)/(1-sin(z)): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 10 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 10 do seq(coeff(P[n], x, j), j = 0 .. n) end do; # yields sequence in triangular form
%t A186363 T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[Exp[-x]/(1 - Sin[x]), {x, 0, n}] n!, T[n - k, 0] Binomial[n, k]];
%t A186363 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 24 2018 *)
%Y A186363 Cf. A186364, A186365.
%K A186363 nonn,tabl
%O A186363 0,8
%A A186363 _Emeric Deutsch_, Feb 28 2011