A186761 - cp's OEIS frontend

A186761 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) . 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.

%I A186761 #12 May 04 2023 18:59:02
%S A186761 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 A186761 1517,1982,833,616,35,56,0,1,12593,14040,9612,2408,1526,56,84,0,1,
%U A186761 111465,134248,72588,35176,5838,3360,84,120,0,1,1122819,1305126,797461,276120,107710,12516,6762,120,165,0,1
%N A186761 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 A186761 Sum of entries in row n is n!.
%C A186761 T(n,0) = A186762(n).
%C A186761 Sum_{k=0..n} k*T(n,k) = A186763(n).
%H A186761 Alois P. Heinz, <a href="/A186761/b186761.txt">Rows n = 0..170, flattened</a>
%F A186761 E.g.f.: G(t,z) = exp((t-1)*sinh z)/(1-z).
%F A186761 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 A186761 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 A186761 T(3,1)=4 because we have (1)(23), (12)(3), (13)(2), and (123).
%e A186761 T(4,1)=4 because we have (1)(243), (143)(2), (142)(3), and (132)(4).
%e A186761 Triangle starts:
%e A186761    1;
%e A186761    0,  1;
%e A186761    1,  0,  1;
%e A186761    1,  4,  0,  1;
%e A186761    9,  4, 10,  0, 1;
%e A186761   33, 56, 10, 20, 0, 1;
%e A186761   ...
%p A186761 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 A186761 # second Maple program:
%p A186761 b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-j)*(
%p A186761       `if`(j::odd, x-1, 0)+(j-1)!)*binomial(n-1, j-1), j=1..n)))
%p A186761     end:
%p A186761 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p A186761 seq(T(n), n=0..14);  # _Alois P. Heinz_, May 12 2017
%t A186761 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 A186761 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
%t A186761 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jun 05 2018, after _Alois P. Heinz_ *)
%Y A186761 Cf. A186762, A186763, A186764, A186766, A186769.
%K A186761 nonn,tabl
%O A186761 0,8
%A A186761 _Emeric Deutsch_, Feb 27 2011

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