A010029 -id:A010029 - cp's OEIS frontend

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

Emeric Deutsch and Vladeta Jovovic, Dec 17 2007

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).

			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;
  ...

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

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

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

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 *)

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

A010030 Irregular triangle read by rows: T(n,k) (n >= 1, 0 <= k <= [n/2]) = number of permutations of 1..n with [n/2]-k runs of consecutive pairs up and down (divided by 2).

Original entry on oeis.org

1, 1, 0, 3, 0, 3, 8, 1, 25, 28, 7, 17, 155, 143, 45, 259, 1005, 933, 323, 131, 2770, 7488, 7150, 2621, 3177, 27978, 64164, 62310, 23811, 1281, 51433, 294602, 619986, 607445, 239653

Offset: 1

N. J. A. Sloane

			Triangle begins:
1,
1, 0,
3, 0,
3, 8, 1,
25, 28, 7,
17, 155, 143, 45,
259, 1005, 933, 323,
131, 2770, 7488, 7150, 2621,
3177, 27978, 64164, 62310, 23811,
1281, 51433, 294602, 619986, 607445, 239653,
...

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

Cf. A002464, A001266, A000239, A000544, A001282.

G.f. for number of permutations of 1..n by number of runs of consecutive pairs up and down is Sum(n!*(((1-y)*(2*x^2-x^3)-x)/((1-y)*x^2-1))^n,n = 0 .. infinity), cf. A010029. - Vladeta Jovovic, Nov 23 2007

More terms from Vladeta Jovovic, Nov 23 2007

Entry revised by N. J. A. Sloane, Apr 14 2014

cp's OEIS Frontend

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)).

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