A134830 Triangle of rank k of permutations of {1,2,...,n}.

1, 1, 0, 1, 0, 1, 2, 1, 1, 2, 6, 4, 3, 2, 9, 24, 18, 14, 11, 9, 44, 120, 96, 78, 64, 53, 44, 265, 720, 600, 504, 426, 362, 309, 265, 1854, 5040, 4320, 3720, 3216, 2790, 2428, 2119, 1854, 14833, 40320, 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy
• tabl

Comments

The rank k of a permutation of n elements is the first position of a fixed point. If there is no fixed point then k=n+1 and R(n,n+1)=A000166(n), the derangements numbers (subfactorials).

Examples

			Triangle begins:
  [1];
  [1,0];
  [1,0,1];
  [2,1,1,2];
  [6,4,3,2,9];
  [24,18,14,11,9,44];
  ...
R(4,2)=4 from the four rank k=2 partitions of 4 elements (3,2,1,4), (3,2,4,1), (4,2,1,3) and (4,2,3,1).
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 176, Table 5.3 (without row n=0 and column k=1), p. 185.

Links

• Wolfdieter Lang, First 10 rows and more.

Crossrefs

Row sums n!=A000142(n). Alternating row sums A134831.

Formula

R(n,k)=0 if n+1=0, with D(n):=A000166(n) the derangements numbers (subfactorials), R(n,k)=sum((-1)^j*binomial(k-1,j)*(n-j-1)!,j=0..k-1), k from 1,..,n.

Subtriangle without diagonal k=n+1: R(n,k)=sum(binomial(n-k,j)*D(k+j-1),j=0..n-k), k=1,...,n, n>=1, with D(n):=A000166(n).

R(n,k) = R(n,k-1) - R(n-1,k-1), R(0,0)=1, R(n,1)=(n-1)!.