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.

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

Author

N. J. A. Sloane

Keywords

• tabf
• nonn
• nice

Examples

			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 ...
  ...
		

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

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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