A003322 Necklace permutations. (Formerly M1282)

1, 1, 1, 2, 4, 14, 57, 347, 2375, 20752, 200805, 2293192, 28097136, 381963696, 5651257984, 87997162749, 1478237810536, 26259254617337, 491527066667949, 9653395554710512, 200955114102582726, 4317192207291540985, 97857028247700646166, 2306560514881759970113

Offset

1,4

Comments

a(n) counts distinct sequences of beads chosen to be replaced, up to rotation and reflection before the replacement. A397015 instead counts distinct sequences of necklaces after replacement. The current sequence is larger at n=9, 12, 15, 16, ... because it is possible for distinct beads to be chosen and result in the same necklace. E.g. BBWBWWBWW has 5 distinct white beads but only 4 resulting necklaces; replacing the 2nd white bead creates BBWBBWBWW, whereas replacing the 4th white bead creates BBWBWWBBW which is equivalent after rotation. - D. S. McNeil and Martin Fuller, Jun 13 2026

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Martin Fuller, Table of n, a(n) for n = 1..35

H. W. Gould, Letters to N. J. A. Sloane, 1974.

D. E. Knuth, Computer science and its relation to mathematics, Amer. Math. Monthly, 81 (1974), 323-343. doi: 10.2307/2318994.

D. E. Knuth, H. Wilf, C. L. Mallows, & D. Klarner, Correspondence, 1994,

Index entries for sequences related to necklaces.

Crossrefs

Cf. A397015.

Sequence in context: A006384 A059687 A204198 * A170939 A030952 A030861

Adjacent sequences: A003319 A003320 A003321 * A003323 A003324 A003325

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

Corrected and extended by Don Knuth, Nov 15 1995

a(20) moved to A397015 and replaced by D. S. McNeil and Martin Fuller, Jun 13 2026

a(21) onward from Martin Fuller, Jun 13 2026

Status

approved