A268559 Number of bicolored immersions of unoriented circle into oriented surface with n double points, up to stable geotopy.

1, 2, 3, 14, 54, 420, 3886, 46470, 645524, 10328214, 185800748, 3716014090, 81749732156, 1961993502732, 51011757342810, 1428329205659176, 42849873773205576, 1371195960742379536, 46620662578043190450, 1678343852809487646738, 63777066403240838651416

Offset

0,2

Comments

a(n) is also the number of generic directed embeddings of n-arc directed bouquets (see Links for the details). - Andrei Zabolotskii, Jan 20 2026

Links

Table of n, a(n) for n=0..20.

Robert Coquereaux and Jean-Bernard Zuber, Maps, immersions and permutations, Journal of Knot Theory and Its Ramifications, Vol. 25, No. 8 (2016), 1650047; arXiv preprint, arXiv:1507.03163 [math.CO], 2015-2016. See Table 9, line UOc total.

Mark N. Ellingham and Joanna A. Ellis-Monaghan, A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries, Symmetry 2022, 14(9), 1793. See (A6).

Andrei Zabolotskii, Immersed curves and embedded bouquets

Formula

a(n) = (A268562(n) + [n is odd] * A098560((n-1)/2)) / 2. [Ellingham and Ellis-Monaghan] - Andrei Zabolotskii, Dec 03 2025

Prog

(SageMath)
def aI(n, k): return sum(euler_phi(n/g) * factorial(g) * (k*n/g)^g for g in divisors(n)) / n
def aR(n, k): return n%2 and factorial(n//2) * 2^(n//2) * k^((n+1)/2)
def a(n): return (aI(n, 2) + aR(n, 2))/2 if n else 1 # Andrei Zabolotskii, Dec 03 2025

Crossrefs

Cf. A268561, A268562.

Sequence in context: A188289 A153741 A070207 * A346057 A371608 A270707

Adjacent sequences: A268556 A268557 A268558 * A268560 A268561 A268562

Keyword

nonn

Author

N. J. A. Sloane, Mar 02 2016

Extensions

Name clarified by Andrei Zabolotskii, Jun 09 2024

a(0) and a(10) onward from Andrei Zabolotskii, Jan 20 2026

Status

approved