A268567 Number of immersions of oriented circle into oriented sphere with n labeled double points.

1, 4, 42, 780, 21552, 803760, 38054160, 2194345440, 149604053760, 11794431720960, 1056927459571200, 106197377365094400, 11831983152533913600, 1448373107312190259200, 193295730632526147225600, 27939600631552720061952000, 4349183288219555957563392000

Offset

1,2

Links

Table of n, a(n) for n=1..17.

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: 10.1142/S0218216516500474.

Formula

a(n) = A054993(n)*(n-1)!/2 [proof: in a long curve, label the first (leftmost) crossing by #1 and the rest by whatever, in (n-1)! ways; then each labeled oriented spherical closed curve corresponds to precisely 2 such labeled long curves, depending on which of the 2 edges going into vertex #1 is chosen to be "long"]. - Andrey Zabolotskiy, Jan 14 2025

Crossrefs

Cf. A054993, A264755, A008986.

Sequence in context: A140055 A221096 A179499 * A197866 A197948 A196194

Adjacent sequences: A268564 A268565 A268566 * A268568 A268569 A268570

Keyword

nonn

Author

N. J. A. Sloane, Mar 03 2016

Extensions

New name and terms a(11) onwards from Andrey Zabolotskiy, Jan 21 2025

Status

approved