A162976 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k double descents and initial descents (n>=0; 0<=k<=max(0,n-1)) [we say that i is a doubledescent of a permutation p if p(i) > p(i+1) > p(i+2); we say that a permutation p has an initial descent if p(1) > p(2)].

1, 1, 1, 1, 3, 2, 1, 9, 11, 3, 1, 39, 48, 28, 4, 1, 189, 297, 166, 62, 5, 1, 1107, 1902, 1419, 476, 129, 6, 1, 7281, 14391, 11637, 5507, 1235, 261, 7, 1, 54351, 118044, 111438, 56400, 19096, 3020, 522, 8, 1, 448821, 1078245, 1119312, 673128, 239146, 61986

Offset: 0

Author

Emeric Deutsch, Jul 26 2009

Keywords

• nonn
• tabf

Comments

Sum of entries in row n is n! = A000142(n).

T(n,0) = A080635(n).

Examples

			T(4,2) = 3 because each of the permutations 4312, 4213, and 3214 has one doubledescent and one initial descent.
Triangle starts:
:   1;
:   1;
:   1,   1;
:   3,   2,   1;
:   9,  11,   3,  1;
:  39,  48,  28,  4, 1;
: 189, 297, 166, 62, 5, 1;
		

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983 (p. 195).

Links

• Alois P. Heinz, Rows n = 0..141, flattened
• Yan Zhuang, Monoid networks and counting permutations by runs, arXiv preprint, 2015.
• Y. Zhuang, Counting permutations by runs, J. Comb. Theory Ser. A 142 (2016), pp. 147-176.

Crossrefs

Cf. A000142, A080635, A162975.

Programs

Maple

eq := s^2-(t+1)*s+1 = 0: sol := solve(eq, s): a := sol[1]: b := sol[2]: G := (exp(b*z)-exp(a*z))/(b*exp(a*z)-a*exp(b*z)): Gser := simplify(series(G, z = 0, 15)): P[0]:=1: for n to 11 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 11 do seq(coeff(P[n], t, j), j = 0 .. max(0,n-1)) end do;
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, 1), j=1..u)+
      add(b(u+j-1, o-j, 2)*`if`(t=2, x, 1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Dec 09 2016

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, 2]*If[t == 2, x, 1], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}] ][b[0, n, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

E.g.f.: G(t,z) = [exp(bz)-exp(az)]/[b*exp(az)-a*exp(bz)], where a+b=1+t and ab=1.

Extensions

One term for row n=0 prepended by Alois P. Heinz, Dec 09 2016