A001280 -id:A001280 - 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

A010029 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= floor(n/2)) = number of permutations of 1..n with exactly floor(n/2) - k runs of consecutive pairs up.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			Triangle begins
     1
     1     1
     3     3
     1    12    11
    11    56    53
     3    87   321    309
    53   693  2175   2119
    11   680  5934  17008   16687
   309  8064 55674 150504  148329
    53  5805 96370 572650 1485465 1468457
  2119 95575 ...
  ...

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

Vincenzo Librandi, Rows n = 1..50, flattened

Cf. A000255, A001277, A001278, A001279, A001280, A000142 (row sums), A136123 (rows reversed).

A010029 := 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,floor(n/2)-k) ;
end proc:
seq(seq( A010029(n,k),k=0..floor(n/2)),n=1..12) ; # R. J. Mathar, Jul 01 2022

max = 16; coes = CoefficientList[ Series[ Sum[ n!*(((1 - y)*x^2 - x)/((1 - y)*x^2 - 1))^n, {n, 0, max}], {x, 0, max}, {y, 0, max}], {x, y}]; Table[ Table[ coes[[n, k]] , {k, 1, Floor[(n + 1)/2]}] // Reverse, {n, 2, max - 4}] // Flatten (* Jean-François Alcover, Jan 10 2013, after Vladeta Jovovic *)

G.f.: Sum_{n>=0} n! *( ((1-y)*x^2-x)/((1-y)*x^2-1) )^n, for the triangle read right-to-left. - Vladeta Jovovic, Nov 21 2007

T(n,k) = A136123(n,[n/2]-k). - R. J. Mathar, Jul 01 2022

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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