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

0,6

Comments

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

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.

Example

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

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 *)

Prog

(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

Crossrefs

Column k=0 gives A000296.

Row sums give A000166.

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

Sequence in context: A246495 A050831 A056799 * A259821 A352681 A109503

Adjacent sequences: A216960 A216961 A216962 * A216964 A216965 A216966

Keyword

nonn,tabf

Author

N. J. A. Sloane, Sep 27 2012

Extensions

More terms from Michel Marcus, Feb 09 2013

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

Status

approved