A364973 a(n) = number of degree n rational curves in the complex projective plane which satisfy an order 3n-1 local tangency constraint.

1, 1, 4, 26, 217, 2110, 22744, 264057, 3242395, 41596252, 552733376, 7559811021, 105919629403, 1514674166755, 22043665219240, 325734154669786, 4877954835706120, 73914684068584441, 1131820243084746628, 17494508772311055354, 272708269111411142397, 4283702718045699720655

Offset: 1

Author

Kyler Siegel, Aug 22 2023

Keywords

• nonn

Comments

This is the number of degree n rational curves in the complex projective plane which pass through a specified point and have contact order 3n-1 to a generic smooth local divisor through that point.

After multiplying by 3n-1, this sequence appears to agree with A353195.

Computed in terms of Gromov-Witten invariants of rational surfaces in McDuff-Siegel 2021.

Agrees with the count of osculating curves, which are defined and computed by a recursive formula in Muratore 2021.

Computed by a recursive formula in Mikhalkin-Siegel 2023.

Computed by a closed formula as sum over trees with n leaves and combinatorially defined weights in Siegel 2023.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..108
• D. McDuff and K. Siegel, Counting curves with local tangency constraints, J. Top., 14 (2021) 1176-1242.
• G. Mikhalkin and K. Siegel, Ellipsoidal superpotentials and stationary descendants, arXiv:2307.13252 [math.SG] (2023).
• G. Muratore, A recursive formula for osculating curves, Arkiv Mat., 59 (2021) 195-211.
• K. Siegel, Computing higher symplectic capacities I, Int. Math. Res. Not., 2022 (2021) 12402-12461.
• K. Siegel, A tree formula for the ellipsoidal superpotential of the complex projective plane (2023).

Crossrefs

Cf. A353195 (after multiplying by 3n-1).

Programs

Python

from functools import lru_cache
from fractions import Fraction
from math import prod, factorial
from sympy.utilities.iterables import partitions
@lru_cache(maxsize=None)
def A364973(n): # after Sage code
    return 1 if n==1 else int(factorial(3*n-2)*(Fraction(1,factorial(n)**3)-sum(Fraction(prod(((3*m-1)*A364973(m))**e for m, e in p.items()),factorial(3*n-s)*prod(factorial(c) for c in p.values())) for s, p in partitions(n, k=n-1, size=True)))) # Chai Wah Wu, Nov 10 2023

Sage

# The following code is written in Sage and is based on the recursive formula in Mikhalkin-Siegel 2023. Note that a slightly more efficient formula is given by using Partitions instead of OrderedPartitions, at the cost of an extra combinatorial factor.
def a(n):
	if n == 1:
		return 1
	out = 1/(factorial(n)**3)
	for p in [p for p in OrderedPartitions(n) if len(p) > 1]:
		k = len(p)
		summand = prod([(3*m-1)*a(m) for m in p])
		summand /= factorial(k)*factorial(3*n-k)
		out -= summand
	out *= factorial(3*n-2)
	return out
for n in range(1,20):
	print(a(n))

Extensions

More terms from Chai Wah Wu, Nov 10 2023