A180193 Triangle read by rows: T(n,k) is the number of permutations of [n] having k blocks of odd length (0<=k<=n).

1, 0, 1, 1, 0, 1, 0, 3, 0, 3, 2, 0, 11, 0, 11, 0, 14, 0, 53, 0, 53, 6, 0, 96, 0, 309, 0, 309, 0, 78, 0, 724, 0, 2119, 0, 2119, 24, 0, 852, 0, 6070, 0, 16687, 0, 16687, 0, 504, 0, 9300, 0, 56418, 0, 148329, 0, 148329, 120, 0, 8040, 0, 106170, 0, 577556, 0, 1468457, 0

Offset: 0

Author

Emeric Deutsch, Sep 09 2010

Keywords

• nonn
• tabl

Comments

A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67.

Examples

			T(3,1)=3 because we have (123), 23(1), and (3)12 (the blocks of odd length are shown between parentheses). T(4,0)=2 because we have 1234 and 3412.
Triangle starts:
  1;
  0,1;
  1,0,1;
  0,3,0,3;
  2,0,11,0,11;
  0,14,0,53,0,53;
		

Links

• A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.

Crossrefs

Cf. A000166, A000255, A001564, A180194, A180195

Programs

Maple

d[ -1] := 0: d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if `mod`(n+k, 2) = 1 then 0 else sum(binomial(k+j, j)*binomial((1/2)*n+(1/2)*k-1, k+j-1)*(d[k+j]+d[k+j-1]), j = 0 .. (1/2)*n-(1/2)*k) end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

Formula

T(n,k) = Sum(binomial(k+j,j)*binomial((n+k+2)/2,k+j-1)*[d(k+j)+d(k+j-1)], j=0..(n-k)/2) if n and k are of the same parity; T(n,k)=0 if n and k have opposite parities (0<=k<=n).

T(n,n) = T(n,n-2) = d(n)+d(n-1) = A000255(n-1), where d(i)=A000166(i) are the derangement numbers.

T(2n+1,1) = A001564(n).

Sum(k*T(n,k),k>=0) = A180195(n).

Sum of entries in row n = n! = A000142(n).