A216963 Triangle read by rows, arising in enumeration of permutations by cyclic peaks, cycles and fixed points.

1, 0, 1, 1, 1, 4, 5, 11, 28, 5, 41, 153, 71, 162, 872, 759, 61, 715, 5191, 7262, 1665, 3425, 32398, 66510, 29778, 1385, 17722, 211937, 601080, 443231, 60991, 98253, 1451599, 5446847, 5994473, 1642877, 50521, 580317, 10393114, 49940615, 76889330, 35162440, 3249025

Offset: 0

Author

N. J. A. Sloane, Sep 27 2012

Keywords

• nonn
• tabf

Comments

See Ma and Chow (2012) for precise definition (cf. Proposition 3).

Examples

			Triangle begins:
:   1;
:   0;
:   1;
:   1,    1;
:   4,    5;
:  11,   28,    5;
:  41,  153,   71;
: 162,  872,  759,   61;
: 715, 5191, 7262, 1665;
...
		

Links

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

Column k=0 gives A000296.

Row sums give A000166.

T(2n+1,n) gives A000364(n) for n>0.

Programs

Maple

p:= proc(n) option remember; expand(`if`(n<4,
      [1, 0, x, x*(1+q)][n+1], (n-1)*q*p(n-1)+
      2*q*(1-q)*diff(p(n-1), q)+x*(1-q)*
      diff(p(n-1), x)+(n-1)*x*p(n-2)))
    end:
T:= n-> (t-> seq(coeff(t, q, i), i=0..
         max(0, degree(t))))(subs(x=1, p(n))):
seq(T(n), n=0..15);  # Alois P. Heinz, Apr 13 2017

Mathematica

p[0] = 1; p[1] = 0; p[2] = x; p[3] = (1 + q) x;
p[n_] := p[n] = Expand[(n - 1) q p[n - 1] + 2 q (1 - q) D[p[n - 1], q] + x (1 - q) D[p[n - 1], x] + (n - 1) x p[n - 2]];
T[n_] := CoefficientList[p[n] /. x -> 1 , q]; T[1] = {0};
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Nov 08 2017 *)

PARI

tabf(m) = {P = x; M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); Q = (1+q)*x; M = subst(Q, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); for (n=3, m, newP = n*q*Q + 2*q*(1-q)*deriv(Q,q) + x*(1-q)*deriv(Q,x) + n*x*P; M = subst(newP, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = Q; Q = newP;);} \\ Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013

One row for T(0,0)=1 prepended by Alois P. Heinz, Apr 13 2017