A216964 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.

1, 2, 6, 22, 2, 94, 26, 460, 244, 16, 2532, 2124, 384, 15420, 18536, 6092, 272, 102620, 166440, 83436, 10384, 739512, 1550864, 1082712, 247776, 7936, 5729192, 15040112, 13841928, 4864480, 441088, 47429896, 151960264, 177512632, 87003032, 14741984, 353792

Offset

1,2

Comments

See Ma and Chow (2012) for precise definition (see Corollary 5).

Links

Table of n, a(n) for n=1..37.

Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.

Example

Triangle begins:
1
2
6
22, 2
94, 26
460, 244, 16
2532, 2124, 384
...

Mathematica

rows = 12;
Reap[For[P = x*y; n = 1; Sow[{1}], n < rows, n++, P = (n*q + x*y)*P + 2*q*(1-q)*D[P, q] + 2*x*(1-q)*D[P, x] + (1-2*y+q*y)*D[P, y] // Simplify; Sow[CoefficientList[P /. {x -> 1, y -> 1}, q]]]][[2, 1]] // Flatten (* Jean-François Alcover, Sep 23 2018, from PARI *)

Prog

(PARI) tabf(m) = {P = x*y; for (n=1, m, M = subst(P, x, 1); M = subst(M, y, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = (n*q+x*y)*P + 2*q*(1-q)*deriv(P, q)+ 2*x*(1-q)*deriv(P, x)+ (1-2*y+q*y)*deriv(P, y); ); } \\ Michel Marcus, Feb 08 2013

Crossrefs

First column is A187251.

Sequence in context: A263486 A182544 A216120 * A187250 A129534 A216719

Adjacent sequences: A216961 A216962 A216963 * A216965 A216966 A216967

Keyword

nonn,tabf

Author

N. J. A. Sloane, Sep 27 2012

Extensions

More terms from Michel Marcus, Feb 08 2013

Status

approved