A067601 a(n) is the number of inequivalent permutations of {0..2n-1}, such that the first differences (modulo 2n) are a permutation of {1..2n-1}.

1, 1, 2, 12, 144, 1928, 44664, 1377984, 51826560

Offset: 1

Author

Eugene McDonnell (eemcd(AT)aol.com), Jan 31 2002

Keywords

• nonn
• more

Comments

"Inequivalent" effectively means that the permutation begins with 0 and the second item is <= n. (Working modulo 2n, s1+k,s2+k,s3+k,... is equivalent to s1,s2,s3,...; and -s1,-s2,-s3 is equivalent to s1,s2,s3,...)

The references all deal with length 12.

Examples

			0 1 3 2 has first difference, mod 4, of 1 2 3;
0 2 1 4 5 3 has first difference, mod 6, of 2 5 3 1 4;
0 4 5 8 3 1 7 9 2 11 10 6 has first difference, mod 12, of 4 1 3 7 10 6 2 5 9 11 8.
		

Links

• Stefan Bauer-Mengelberg and Melvin Ferentz, On Eleven-Interval Twelve-Tone Rows, Perspectives of New Music 3, no. 2 (Spring-Summer 1965): 93-103
• Sean A. Irvine, Java program (github)
• Robert Morris and Daniel Starr, The Structure of All-interval Series, Journal of Music Theory 18, no. 2 (Fall 1974): 364-389.
• David Schiff, Elliott Carter's Harvest Home, Tempo 167 (December 1988): 7-13.

Crossrefs

Cf. A141598, A141599.

Formula

a(n) = ceiling(A141599(n)/2). - Leo C. Stein, Nov 26 2016

Extensions

Edited by Don Reble, Oct 31 2005

a(9) from Sean A. Irvine, Dec 22 2023