A186764 - cp's OEIS frontend

A186764 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing even 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) . A cycle is said to be even if it has an even number of entries. For example, the permutation (18)(2347)(569) has 2 increasing even cycles.

%I A186764 #14 Feb 05 2025 14:46:48
%S A186764 1,1,1,1,3,3,14,7,3,70,35,15,419,226,60,15,2933,1582,420,105,23421,
%T A186764 12741,3423,630,105,210789,114669,30807,5670,945,2108144,1144921,
%U A186764 311160,55755,7875,945,23189584,12594131,3422760,613305,86625,10395,278279165,151125052,41041968,7429290,1001385,114345,10395
%N A186764 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing even 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)<... .  A cycle is said to be even if it has an even number of entries. For example, the permutation (18)(2347)(569) has 2 increasing even cycles.
%C A186764 Row n contains 1+floor(n/2) entries.
%C A186764 Sum of entries in row n is n!.
%C A186764 T(n,0) = A186765(n).
%C A186764 Sum(k*T(n,k), k>=0) = A080227(n).
%H A186764 Alois P. Heinz, <a href="/A186764/b186764.txt">Rows n = 0..200, flattened</a>
%F A186764 E.g.f.: G(t,z) = exp((t-1)(cosh z - 1))/(1-z).
%F A186764 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 A186764 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 A186764 We have: G(t,z) = H(1,t,1,1,z).
%e A186764 T(3,1)=3 because we have (1)(23), (12)(3), and (13)(2).
%e A186764 T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
%e A186764 Triangle starts:
%e A186764    1;
%e A186764    1;
%e A186764    1,  1;
%e A186764    3,  3;
%e A186764   14,  7,  3;
%e A186764   70, 35, 15;
%e A186764   ...
%p A186764 g := exp((t-1)*(cosh(z)-1))/(1-z): gser := simplify(series(g, z = 0, 15)): for n from 0 to 12 do P[n] := sort(expand(factorial(n)*coeff(gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%p A186764 # second Maple program:
%p A186764 b:= proc(n) option remember; expand(
%p A186764       `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*
%p A186764       `if`(j::odd, (j-1)!, x+((j-1)!-1)), j=1..n)))
%p A186764     end:
%p A186764 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186764 seq(T(n), n=0..14);  # _Alois P. Heinz_, Apr 13 2017
%t A186764 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1]*If[ OddQ[j], (j-1)!, x+(j-1)!-1], {j, 1, n}]]];
%t A186764 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];
%t A186764 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, May 03 2017, after _Alois P. Heinz_ *)
%Y A186764 Cf. A186761, A186765, A186766, A186769, A080227.
%K A186764 nonn,tabf
%O A186764 0,5
%A A186764 _Emeric Deutsch_, Feb 27 2011

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