A136123 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k maximal strings of increasing consecutive integers (0<=k<=floor(n/2)).

1, 1, 1, 1, 3, 3, 11, 12, 1, 53, 56, 11, 309, 321, 87, 3, 2119, 2175, 693, 53, 16687, 17008, 5934, 680, 11, 148329, 150504, 55674, 8064, 309, 1468457, 1485465, 572650, 96370, 5805, 53, 16019531, 16170035, 6429470, 1200070, 95575, 2119

Offset

0,5

Comments

Row n has 1+floor(n/2) terms. Row sums are the factorials (A000142). Column 0 yields A000255. Column 1 yields A001277. Column 2 yields A001278. Column 3 yields A001279. Column 4 yields A001280. Sum(k*T(n,k),k>=0)=(n-2)!*(n^2 - 3n + 3)=A001564(n-2).

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.

Links

Table of n, a(n) for n=0..41.

Formula

G.f.: G(x,t) = Sum_{n>=0} n!*((1-t)*x^2 - x)/((1-t)*x^2-1))^n. - Vladeta Jovovic

Example

T(3,0)=3 because we have 132, 213 and 321; T(6,3)=3 because we have 125634, 341256, 563412.
Triangle starts:
    1;
    1;
    1,   1;
    3,   3;
   11,  12,  1;
   53,  56, 11;
  309, 321, 87, 3;
  ...

Maple

G:=Sum(factorial(n)*(((1-t)*x^2-x)/((1-t)*x^2-1))^n, n=0..infinity): Gser:= simplify(series(G, x=0, 13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, x, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
# alternative
A136123 := proc(n, k)
    add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!, i=0..n+1) ;
    coeftayl(%, x=0, n) ;
    coeftayl(%, y=0, k) ;
end proc:
seq(seq( A136123(n, k), k=0..floor(n/2)), n=0..12) ; # R. J. Mathar, Jul 01 2022

Mathematica

T[n_, k_] := Sum[x^i*(((1-y)*x-1)/((1-y)*x^2-1))^i*i!, {i, 0, n+1}] //
   SeriesCoefficient[#, {x, 0, n}]& //
   SeriesCoefficient[#, {y, 0, k}]&;
Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 09 2023, after R. J. Mathar *)

Crossrefs

Cf. A000142, A000255, A001277, A001278, A001279, A001280, A001564, A010029 (rows reversed).

Sequence in context: A146458 A122573 A230011 * A045495 A045494 A217712

Adjacent sequences: A136120 A136121 A136122 * A136124 A136125 A136126

Keyword

nonn,tabf

Author

Emeric Deutsch and Vladeta Jovovic, Dec 17 2007

Status

approved