A186766 - cp's OEIS frontend

A186766 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing odd cycles (0<=k<=floor(n/3)). 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 1 nonincreasing odd cycle.

%I A186766 #10 May 03 2017 07:51:13
%S A186766 1,1,2,5,1,20,4,77,43,472,238,10,2585,2385,70,21968,16504,1848,157113,
%T A186766 189695,15792,280,1724064,1591082,310854,2800,15229645,21449481,
%U A186766 3100614,137060,204738624,213397204,59267252,1583120,15400,2151199429,3347368503,676271024,51981644,200200
%N A186766 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing odd cycles (0<=k<=floor(n/3)). 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 1 nonincreasing odd cycle.
%C A186766 Row n has 1+floor(n/3) entries.
%C A186766 Sum of entries in row n is n!.
%C A186766 T(n,0) = A186767(n).
%C A186766 Sum(k*T(n,k),k>=0) = A186768(n).
%H A186766 Alois P. Heinz, <a href="/A186766/b186766.txt">Rows n = 0..200, flattened</a>
%F A186766 E.g.f.: G(t,z)=exp((1-t)sinh z)*(1+z)^{(t-1)/2}/(1-z)^{(t+1)/2}.
%F A186766 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 A186766 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 A186766 We have: G(t,z)=H(1,1,t,1,z).
%e A186766 T(3,1)=1 because we have (132).
%e A186766 T(4,1)=4 because we have (1)(243), (143)(2), (142)(3), and (132)(4).
%e A186766 Triangle starts:
%e A186766 1;
%e A186766 1;
%e A186766 2;
%e A186766 5,1;
%e A186766 20,4;
%e A186766 77,43;
%p A186766 g := exp((1-t)*sinh(z))*(1+z)^((t-1)*1/2)/(1-z)^((t+1)*1/2): gser := simplify(series(g, z = 0, 16)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%p A186766 # second Maple program:
%p A186766 b:= proc(n) option remember; expand(
%p A186766       `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*
%p A186766       `if`(j::even, (j-1)!, 1+x*((j-1)!-1)), j=1..n)))
%p A186766     end:
%p A186766 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186766 seq(T(n), n=0..14);  # _Alois P. Heinz_, Apr 13 2017
%t A186766 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1]*If[ EvenQ[j], (j-1)!, 1+x*((j-1)!-1)], {j, 1, n}]]];
%t A186766 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];
%t A186766 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, May 03 2017, after _Alois P. Heinz_ *)
%Y A186766 Cf. A186761, A186764, A186767, A186768, A186769.
%K A186766 nonn,tabf
%O A186766 0,3
%A A186766 _Emeric Deutsch_, Feb 27 2011

cp's OEIS Frontend