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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. 11
1, 1, 1, 1, 3, 3, 14, 7, 3, 70, 35, 15, 419, 226, 60, 15, 2933, 1582, 420, 105, 23421, 12741, 3423, 630, 105, 210789, 114669, 30807, 5670, 945, 2108144, 1144921, 311160, 55755, 7875, 945, 23189584, 12594131, 3422760, 613305, 86625, 10395, 278279165, 151125052, 41041968, 7429290, 1001385, 114345, 10395 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n contains 1+floor(n/2) entries.

Sum of entries in row n is n!.

T(n,0) = A186765(n).

Sum(k*T(n,k), k>=0) = A080227(n).

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

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

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

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}.

We have: G(t,z) = H(1,t,1,1,z).

EXAMPLE

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

T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).

Triangle starts:

1;

1;

1,1;

3,3;

14,7,3;

70,35,15;

MAPLE

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

# second Maple program:

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

      `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*

      `if`(j::odd, (j-1)!, x+((j-1)!-1)), j=1..n)))

    end:

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

seq(T(n), n=0..14);  # Alois P. Heinz, Apr 13 2017

MATHEMATICA

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[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];

Table[T[n], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, May 03 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A186761, A186765, A186766, A186769, A080227.

Sequence in context: A288294 A288803 A288334 * A204767 A287904 A287946

Adjacent sequences:  A186761 A186762 A186763 * A186765 A186766 A186767

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Feb 27 2011

STATUS

approved

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.