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A186756 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing 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) < ... .

%I A186756 #20 May 11 2025 23:48:45
%S A186756 1,1,2,5,1,15,9,52,68,203,507,10,877,3918,245,4140,32057,4123,21147,
%T A186756 280700,60753,280,115975,2645611,853914,13300,678570,26917867,
%U A186756 11923428,396935,4213597,295934526,169127222,9710855,15400,27644437,3513447546,2469452843,215274774,1201200
%N A186756 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k nonincreasing 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) < ... .
%C A186756 Row n contains 1 + floor(n/3) entries.
%H A186756 Alois P. Heinz, <a href="/A186756/b186756.txt">Rows n = 0..250, flattened</a>
%F A186756 E.g.f.: G(t,z) = exp((1-t)*(exp(z)-1))/(1-z)^t.
%F A186756 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(1,1,t,z).
%F A186756 Sum_{k=0..n} T(n,k) = n!.
%F A186756 T(n,0) = A000110(n) (the Bell numbers).
%F A186756 Sum_{k=0..n} k*T(n,k) = A121633(n).
%e A186756 T(3,0) = 5 because we have (1)(2)(3), (1)(23), (12)(3), (13)(2), and (123).
%e A186756 T(3,1) = 1 because we have (132).
%e A186756 T(4,1) = 9 because we have (1)(243), (1432), (142)(3), (132)(4), (1342), (1423), (1243), (143)(2), and (1324).
%e A186756 Triangle starts:
%e A186756     1;
%e A186756     1;
%e A186756     2;
%e A186756     5,   1;
%e A186756    15,   9;
%e A186756    52,  68;
%e A186756   203, 507, 10;
%p A186756 G := exp((1-t)*(exp(z)-1))/(1-z)^t: 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, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%p A186756 # second Maple program:
%p A186756 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A186756       (1+x*(i!-1))*b(n-i-1)*binomial(n-1, i), i=0..n-1)))
%p A186756     end:
%p A186756 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186756 seq(T(n), n=0..15);  # _Alois P. Heinz_, Sep 26 2016
%t A186756 b[n_] := b[n] = Expand[If[n==0, 1, Sum[(1+x*(i!-1))*b[n-i-1]*Binomial[n-1, i], {i, 0, n-1}]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Nov 28 2016 after _Alois P. Heinz_ *)
%Y A186756 Cf. A000110, A121633, A186754, A186755, A186757, A186758, A186759, A186760.
%K A186756 nonn,tabf
%O A186756 0,3
%A A186756 _Emeric Deutsch_, Feb 26 2011