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A186759 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) < ... .

%I A186759 #24 May 11 2025 23:48:51
%S A186759 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 A186759 162,1804,1904,903,210,56,0,1,715,12971,15727,8008,2408,406,84,0,1,
%U A186759 3425,104600,142533,80323,25662,5502,714,120,0,1,17722,936370,1418444,871575,303385,68712,11256,1170,165,0,1
%N A186759 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) < ... .
%H A186759 Alois P. Heinz, <a href="/A186759/b186759.txt">Rows n = 0..140, flattened</a>
%F A186759 E.g.f.: G(t,z) = exp((1-t)*(exp(z)-1-z))/(1-z)^t.
%F A186759 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(u*z+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).
%F A186759 Sum_{k=0..n} T(n,k) = n!.
%F A186759 T(n,0) = A000296(n).
%F A186759 Sum_{k=0..n} k*T(n,k) = A186760(n).
%e A186759 T(3,1) = 4 because we have (1)(23), (12)(3), (13)(2), and (132).
%e A186759 T(4,4) = 1 because we have (1)(2)(3)(4).
%e A186759 Triangle starts:
%e A186759    1;
%e A186759    0,  1;
%e A186759    1,  0,  1;
%e A186759    1,  4,  0,  1;
%e A186759    4,  9, 10,  0, 1;
%e A186759   11, 53, 35, 20, 0, 1;
%p A186759 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 A186759 # second Maple program:
%p A186759 b:= proc(n) option remember; expand(
%p A186759       `if`(n=0, 1, add(b(n-i)*binomial(n-1, i-1)*
%p A186759       `if`(i=1, x, 1+x*((i-1)!-1)), i=1..n)))
%p A186759     end:
%p A186759 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186759 seq(T(n), n=0..10);  # _Alois P. Heinz_, Sep 25 2016
%t A186759 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 A186759 Cf. A000296, A186754, A186755, A186756, A186757, A186758, A186760.
%K A186759 nonn,tabl
%O A186759 0,8
%A A186759 _Emeric Deutsch_, Feb 26 2011