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feat(Combinatorics): Asymptotics of the number of integer partitions #210

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aa9723a
hardy-ramanujan formula
May 11, 2026
c6c0af0
Merge branch 'main' into partition
thomaskwaring May 11, 2026
074bec2
add informal solution
thomaskwaring May 11, 2026
4ffd504
add import
May 12, 2026
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20 changes: 20 additions & 0 deletions LeanEval/Combinatorics/PartitionAsymptotics.lean
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import EvalTools.Markers
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Combinatorics.Enumerative.Partition.Basic
import Mathlib.Analysis.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

/-! # Asymptotic expression for the number of integer partitions

Hardy and Ramanujan's asymptotic expression for the number of integer partitions.

G. H. Hardy, S. Ramanujan. *Asymptotic formulae in combinatory analysis*, 1918.
-/

open Asymptotics Filter Fintype

/-- Hardy and Ramanujan's asymptotic formula. -/
@[eval_problem]
theorem isEquivalent_card_partition :
(fun (n : ℕ) => (card n.Partition : ℝ)) ~[atTop]
(fun n => Real.exp (Real.pi * Real.sqrt (2 * n / 3)) / (4 * n * Real.sqrt 3)) := sorry
10 changes: 10 additions & 0 deletions manifests/problems.toml
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Expand Up @@ -646,6 +646,16 @@ holes = ["neukirch_uchida"]
submitter = "Junyan Xu"
source = "Jürgen Neukirch, Alexander Schmidt, Kay Wingberg. *Cohomology of Number Fields*, Theorem 12.2.1."

[[problem]]
id = "asymptotic_partition"
title = "Hardy-Ramanujan asymptotic formula for the number of integer partitions"
test = false
module = "LeanEval.Combinatorics.PartitionAsymptotics"
holes = ["isEquivalent_card_partition"]
submitter = "Thomas K Waring"
source = "G. H. Hardy, S. Ramanujan. *Asymptotic formulae in combinatory analysis*, 1918."
informal_solution = "Evaluate the growth-rate of coefficients in the generating function by the so-called circle method."

[[problem]]
id = "balanceable_bounded_partitions"
title = "Balanceable k-bounded partitions"
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