A094315 Triangle read by rows giving number of circular permutations of n letters such that all letters are displaced by no more than k places from their original position.

1, 0, 1, 0, 0, 2, 0, 0, 0, 6, 1, 0, 6, 8, 9, 2, 15, 20, 40, 30, 13, 20, 72, 180, 176, 180, 72, 20, 144, 609, 1106, 1421, 980, 595, 154, 31, 1265, 4960, 9292, 10352, 8326, 4096, 1676, 304, 49

Offset: 0

Author

N. J. A. Sloane, based on a suggestion from Anthony C Robin, Jun 02 2004

Keywords

• nonn
• tabl

Comments

The n-th row sums to n!.

Examples

			1;
0, 1;
0, 0, 2;
0, 0, 0, 6;
1, 0, 6, 8, 9;
2, 15, 20, 40, 30, 13;
20, 72, 180, 176, 180, 72, 20;
144, 609, 1106, 1421, 980, 595, 154, 31;
		

References

• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Links

• Anthony C. Robin, 90.72 Circular Wife Swapping, The Mathematical Gazette, Vol. 90, No. 519 (Nov., 2006), pp. 471-478.
• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See Table 2)

Crossrefs

Diagonals give A000183 (which has further references), A000476, A000388, A000380, A000440, etc.

See also A061702.

Formula

Riordan (1954) is devoted to the study of these numbers and gives generating functions (involving the numbers in A061702 as coefficients).