A180185 Triangle read by rows: T(n,k) is the number of permutations of [n] having no 3-sequences and having k successions (0 <= k <= floor(n/2)); a succession of a permutation p is a position i such that p(i +1) - p(i) = 1.

1, 1, 1, 1, 3, 2, 11, 9, 1, 53, 44, 9, 309, 265, 66, 3, 2119, 1854, 530, 44, 16687, 14833, 4635, 530, 11, 148329, 133496, 44499, 6180, 265, 1468457, 1334961, 467236, 74165, 4635, 53, 16019531, 14684570, 5339844, 934472, 74165, 1854, 190899411

Offset: 0

Author

Emeric Deutsch, Sep 06 2010

Keywords

• nonn
• tabf

Comments

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

Sum of entries in row n is A002628(n).

Examples

			T(6,3)=3 because we have 125634, 341256, and 563412.
Triangle starts:
     1;
     1;
     1,    1;
     3,    2;
    11,    9,    1;
    53,   44,    9;
   309,  265,   66,    3;
  2119, 1854,  530,   44;
		

Crossrefs

Cf. A000166, A002628, A002629, A000255.

Programs

Maple

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n, k) if n = 0 and k = 0 then 1 elif k <= (1/2)*n then binomial(n-k, k)*d[n+1-k]/(n-k) else 0 end if end proc: for n from 0 to 12 do seq(a(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form

Mathematica

d[0] = 1; d[n_] := d[n] = n d[n - 1] + (-1)^n;
T[n_, k_] := If[n == 0 && k == 0, 1, If[k <= n/2, Binomial[n - k, k] d[n + 1 - k]/(n - k), 0]];
Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 2]}] // Flatten (* Jean-François Alcover, May 23 2020 *)

PARI

d(n) = if(n<2, !n , round(n!/exp(1)));
for(n=0, 20, for(k=0, (n\2), print1(binomial(n - k, k)*(d(n - k) + d(n - k - 1)),", ");); print();) \\ Indranil Ghosh, Apr 12 2017

Formula

T(n,k) = binomial(n-k,k)*(d(n-k) + d(n-k-1)), where d(j) = A000166(j) are the derangement numbers.

T(n,0) = d(n) + d(n-1) = A000255(n-1).

T(n,1) = d(n).

Sum_{k>=0} k*T(n,k) = A002629(n+1).