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

Author

N. J. A. Sloane, Sep 27 2012

Keywords

• nonn
• tabf

Comments

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

Examples

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

Links

• 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.

Crossrefs

First column is A187251.

Programs

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 *)

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

Extensions

More terms from Michel Marcus, Feb 08 2013