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A186769 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing even cycles (0<=k<=floor(n/4)). 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 even if it has an even number of entries. For example, the permutation (1528)(347)(6) has 1 nonincreasing even cycles.

%I A186769 #10 May 03 2017 07:51:27
%S A186769 1,1,2,6,19,5,95,25,451,269,3157,1883,21092,18353,875,189828,165177,
%T A186769 7875,1660351,1764749,203700,18263861,19412239,2240700,197541565,
%U A186769 237001478,43736682,721875,2568040345,3081019214,568576866,9384375,33029787974,43065489284,10638945317,444068625
%N A186769 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing even cycles (0<=k<=floor(n/4)). 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 even if it has an even number of entries. For example, the permutation (1528)(347)(6) has 1 nonincreasing even cycles.
%C A186769 Row n has 1+floor(n/4) entries.
%C A186769 Sum of entries in row n is n!.
%C A186769 T(n,0) = A186770(n).
%C A186769 Sum(k*T(n,k),k>=0) = A184958(n).
%H A186769 Alois P. Heinz, <a href="/A186769/b186769.txt">Rows n = 0..200, flattened</a>
%F A186769 E.g.f.: G(t,z)=exp((1-t)(cosh z - 1))*(1+z)^{(1-t)/2}/(1-z)^{(1+t)/2}.
%F A186769 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 A186769 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}.
%F A186769 We have: G(t,z) = H(1,1,1,t,z).
%e A186769 T(4,1)=5 because we have (1243), (1324), (1342), (1423), and (1432).
%e A186769 Triangle starts:
%e A186769 1;
%e A186769 1;
%e A186769 2;
%e A186769 6;
%e A186769 19,5;
%e A186769 95,25;
%e A186769 451,269;
%p A186769 g := exp((1-t)*(cosh(z)-1))*(1+z)^((1-t)*1/2)/(1-z)^((1+t)*1/2): gser := simplify(series(g, z = 0, 18)): for n from 0 to 14 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do; for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
%p A186769 # second Maple program:
%p A186769 b:= proc(n) option remember; expand(
%p A186769       `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*
%p A186769       `if`(j::odd, (j-1)!, 1+x*((j-1)!-1)), j=1..n)))
%p A186769     end:
%p A186769 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186769 seq(T(n), n=0..14);  # _Alois P. Heinz_, Apr 13 2017
%t A186769 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-j]*Binomial[n - 1, j - 1]*If[ OddQ[j], (j - 1)!, 1 + x*((j - 1)! - 1)], {j, 1, n}]]];
%t A186769 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];
%t A186769 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, May 03 2017, after _Alois P. Heinz_ *)
%Y A186769 Cf. A186761, A186764, A186766, A186769, A186770, A184958.
%K A186769 nonn,tabf
%O A186769 0,3
%A A186769 _Emeric Deutsch_, Feb 27 2011

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