A008989 Number of immersions of an unoriented circle into the unoriented sphere with n double points.
1, 1, 2, 6, 19, 76, 376, 2194, 14614, 106421, 823832, 6657811, 55557329, 475475046, 4155030702, 36959470662, 333860366236
Offset: 0
Examples
G.f. = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 76*x^5 + 376*x^6 + 2194*x^7 + ...
References
- V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
Links
- J. Cantarella, H. Chapman, and M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. Also Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 40 (2016), DOI: 10.1088/1751-8113/49/40/405001
- 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), 10.1142/S0218216516500474.
- Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math J. (2016) 2:383-405.
Programs
-
Magma
// see A260914
Extensions
a(6)-a(7) from Guy Valette, Feb 09 2004
a(8)-a(9) from Robert Coquereaux and Jean-Bernard Zuber, Jul 21 2015
a(10) from same source added by N. J. A. Sloane, Mar 03 2016
a(11)-a(14) from Brendan McKay, Mar 11 2023
a(15)-a(16) from Brendan McKay, Mar 29 2024