A180186 Triangle read by rows: T(n,k) is the number of permutations of [n] starting with 1, 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, 0, 1, 1, 0, 2, 3, 0, 9, 8, 3, 44, 45, 12, 1, 265, 264, 90, 8, 1854, 1855, 660, 90, 2, 14833, 14832, 5565, 880, 45, 133496, 133497, 51912, 9275, 660, 9, 1334961, 1334960, 533988, 103824, 9275, 264, 14684570, 14684571, 6007320, 1245972, 129780, 5565

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 A165961(n).

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

Sum_{k>=0} k*T(n,k) = A180187(n).

From Emeric Deutsch, Sep 07 2010: (Start)

T(n,k) is also the number of permutations of [n-1] with k fixed points, no two of them adjacent. Example: T(5,2)=3 because we have 1432, 1324, and 3214.

(End)

Examples

			T(5,2)=3 because we have 12453, 12534, and 14523.
Triangle starts:
    1;
    1;
    0,   1;
    1,   0;
    2,   3,   0;
    9,   8,   3;
   44,  45,  12,  1;
  265, 264,  90,  8;
		

Crossrefs

Cf. A000166, A165961, A180187.

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] 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

Formula

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