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A186754 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing 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) < ... .

%I A186754 #30 May 11 2025 23:48:40
%S A186754 1,0,1,0,1,1,1,1,3,1,5,5,7,6,1,23,36,25,25,10,1,129,234,166,110,65,15,
%T A186754 1,894,1597,1316,686,385,140,21,1,7202,12459,10893,5754,2611,1106,266,
%U A186754 28,1,65085,111451,97287,54559,22428,8841,2730,462,36,1,651263,1116277,963121,554670,229405,80871,26397,6000,750,45,1
%N A186754 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing 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) < ... .
%H A186754 Alois P. Heinz, <a href="/A186754/b186754.txt">Rows n = 0..140, flattened</a>
%F A186754 E.g.f.: G(t,z) = exp((t-1)*(exp(z)-1))/(1-z).
%F A186754 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,t,1,z).
%F A186754 Sum_{k=0..n} T(n,k) = n!.
%F A186754 T(n,0) = A186755(n).
%F A186754 Sum_{k=0..n} k*T(n,k) = A002627(n).
%e A186754 T(3,0) = 1 because we have (132).
%e A186754 T(4,2) = 7 because we have (1)(234), (13)(24), (12)(34), (123)(4), (124)(3), (134)(2), and (14)(23).
%e A186754 Triangle starts:
%e A186754    1;
%e A186754    0,  1;
%e A186754    0,  1,  1;
%e A186754    1,  1,  3,  1;
%e A186754    5,  5,  7,  6,  1;
%e A186754   23, 36, 25, 25, 10, 1;
%p A186754 G := exp((t-1)*(exp(z)-1))/(1-z); Gser := simplify(series(G, z = 0, 16)): 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 A186754 # second Maple program:
%p A186754 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A186754       b(n-i)*binomial(n-1, i-1)*(x+(i-1)!-1), i=1..n)))
%p A186754     end:
%p A186754 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186754 seq(T(n), n=0..10);  # _Alois P. Heinz_, Sep 25 2016
%t A186754 b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-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_, Oct 04 2016, after _Alois P. Heinz_ *)
%Y A186754 Columns k=0-1 give: A186755, A349788.
%Y A186754 Cf. A000142, A002627, A186755, A186756, A186757, A186758, A186759, A186760.
%K A186754 nonn,tabl
%O A186754 0,9
%A A186754 _Emeric Deutsch_, Feb 26 2011