A097592 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of even length.

1, 1, 1, 1, 2, 4, 7, 12, 5, 25, 52, 43, 102, 299, 258, 61, 531, 1750, 1853, 906, 3141, 11195, 15634, 8965, 1385, 20218, 83074, 133697, 94398, 31493, 146215, 675304, 1207256, 1088575, 460929, 50521, 1174889, 5880354, 11974457, 12625694, 6632158

Offset: 0

Author

Emeric Deutsch, Aug 29 2004

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/2) entries.

Examples

			Triangle starts:
    1;
    1;
    1,   1;
    2,   4;
    7,  12,   5;
   25,  52,  43;
  102, 299, 258, 61;
Example: T(4,2) = 5 because we have 13/24, 14/23, 23/14, 24/13 and 34/12.
		

Links

• Alois P. Heinz, Rows n = 0..180, flattened

Crossrefs

Columns k=0-10 give: A097597, A317281, A317282, A317283, A317284, A317285, A317286, A317287, A317288, A317289, A317290.

Row sums give A000142.

T(n,floor(n/2)) gives A317139.

T(2n,n) gives A000364.

T(2n+1,n) gives A317140.

Cf. A097591, A097593.

Programs

Maple

G:=2*(t-1)*u/(-2*u+(2-t+t*u)*exp((-1+u)*x/2)+(t-2+t*u)*exp(-(1+u)*x/2)): u:=sqrt(5-4*t): Gser:=simplify(series(G,x=0,12)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(n!*coeff(Gser,x^n)) od: seq(seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)),n=0..11);
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, x^t, expand(
      add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
      add(b(u-j, o+j-1, 0)*x^t, j=1..u)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 19 2013

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, x^t, Expand[Sum[b[u+j-1, o-j, Mod[t+1, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, 0]*x^t, {j, 1, u}]]]; T[n_] := Function[ {p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)

Formula

E.g.f.: 2(t-1)u/[ -2u+(2-t+tu)exp((-1+u)x/2)+(t-2+tu)exp(-(1+u)x/2)], where u=sqrt(5-4t).

Sum_{k=1..floor(n/2)} k * T(n,k) = A097593(n). - Alois P. Heinz, Jul 04 2019