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%I #12 May 04 2023 18:59:02
%S 1,0,1,1,0,1,1,4,0,1,9,4,10,0,1,33,56,10,20,0,1,235,218,211,20,35,0,1,
%T 1517,1982,833,616,35,56,0,1,12593,14040,9612,2408,1526,56,84,0,1,
%U 111465,134248,72588,35176,5838,3360,84,120,0,1,1122819,1305126,797461,276120,107710,12516,6762,120,165,0,1
%N Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... . A cycle is said to be odd if it has an odd number of entries. For example, the permutation (152)(347)(6)(8) has 3 increasing odd cycles.
%C Sum of entries in row n is n!.
%C T(n,0) = A186762(n).
%C Sum_{k=0..n} k*T(n,k) = A186763(n).
%H Alois P. Heinz, <a href="/A186761/b186761.txt">Rows n = 0..170, flattened</a>
%F E.g.f.: G(t,z) = exp((t-1)*sinh z)/(1-z).
%F The 5-variate e.g.f. H(x,y,u,v,z) of permutations with respect to size (marked by z), number of increasing odd cycles (marked by x), number of increasing even cycles (marked by y), number of nonincreasing odd cycles (marked by u), and number of nonincreasing even cycles (marked by v), is given by
%F H(x,y,u,v,z)=exp(((x-u)sinh z + (y-v)(cosh z - 1))*(1+z)^{(u-v)/2}/(1-z)^{(u+v)/2}.
%e T(3,1)=4 because we have (1)(23), (12)(3), (13)(2), and (123).
%e T(4,1)=4 because we have (1)(243), (143)(2), (142)(3), and (132)(4).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 1, 4, 0, 1;
%e 9, 4, 10, 0, 1;
%e 33, 56, 10, 20, 0, 1;
%e ...
%p g := exp((t-1)*sinh(z))/(1-z): gser := simplify(series(g, z = 0, 13)): 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], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-j)*(
%p `if`(j::odd, x-1, 0)+(j-1)!)*binomial(n-1, j-1), j=1..n)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p seq(T(n), n=0..14); # _Alois P. Heinz_, May 12 2017
%t b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*(If[OddQ[j], x - 1, 0] + (j - 1)!)*Binomial[n - 1, j - 1], {j, 1, n}]]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
%t Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jun 05 2018, after _Alois P. Heinz_ *)
%Y Cf. A186762, A186763, A186764, A186766, A186769.
%K nonn,tabl
%O 0,8
%A _Emeric Deutsch_, Feb 27 2011
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