A186759 - OEIS

%I #21 Nov 28 2016 09:06:10

%S 1,0,1,1,0,1,1,4,0,1,4,9,10,0,1,11,53,35,20,0,1,41,280,268,95,35,0,1,

%T 162,1804,1904,903,210,56,0,1,715,12971,15727,8008,2408,406,84,0,1,

%U 3425,104600,142533,80323,25662,5502,714,120,0,1,17722,936370,1418444,871575,303385,68712,11256,1170,165,0,1

%N Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k cycles that are either nonincreasing or of length 1 (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) < ... .

%C Sum of entries in row n is n!.

%C T(n,0) = A000296(n).

%C Sum_{k=0..n} k*T(n,k) = A186760(n).

%H Alois P. Heinz, <a href="/A186759/b186759.txt">Rows n = 0..140, flattened</a>

%F E.g.f.: G(t,z)=exp((1-t)(exp(z)-1-z))/(1-z)^t.

%F The 4-variate e.g.f. H(u,v,w,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of fixed points (marked by u), number of increasing cycles of length >=2 (marked by v), and number of nonincreasing cycles (marked by w) is given by H(u,v,w,z)=exp(uz+v(exp(z)-1-z)+w(1-exp(z))/(1-z)^w. Remark: the nonincreasing cycles are necessarily of length >=3. We have: G(t,z)=H(t,1,t,z).

%e T(3,1)=4 because we have (1)(23), (12)(3), (13)(2), and (132).

%e T(4,4)=1 because we  have (1)(2)(3)(4).

%e Triangle starts:

%e    1;

%e    0,  1;

%e    1,  0,  1;

%e    1,  4,  0,  1;

%e    4,  9, 10,  0, 1;

%e   11, 53, 35, 20, 0, 1;

%p G := exp((1-t)*(exp(z)-1-z))/(1-z)^t: 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, j), j = 0 .. n) end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n) option remember; expand(

%p       `if`(n=0, 1, add(b(n-i)*binomial(n-1, i-1)*

%p       `if`(i=1, x, 1+x*((i-1)!-1)), i=1..n)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Sep 25 2016

%t b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-i]*Binomial[n-1, i-1]*If[i == 1, x, 1+x*((i-1)!-1)], {i, 1, n}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Nov 28 2016 after _Alois P. Heinz_ *)

%Y Cf. A000296, A186754, A186755, A186756, A186757, A186758, A186760.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Feb 26 2011