MacMahon's partition theorem

TheoremAndrews' generalization of MacMahon's partition theorem#✓fully proved

For all natural numbers \(r\) and \(n\), the number of partitions of \(n\) in which every part occurring an odd number of times occurs at least \(2r + 1\) times equals the number of partitions of \(n\) in which every odd part is congruent to \(2r + 1\) modulo \(4r + 2\).

For \(r = 1\) this is MacMahon's partition theorem: the first condition says that no part occurs exactly once, and the second that every part is even or congruent to \(3\) modulo \(6\).

The proof compares generating functions in \(q\). Both sides factor through the product over the even parts, and the remaining factors agree by Euler's theorem that partitions into odd parts and partitions into distinct parts are equinumerous, applied with \(q\) replaced by \(q^{2r+1}\).

public theorem andrews_macmahon_partition_identity
    (r n : ℕ) :
    Set.ncard { p : Nat.Partition n |
      ∀ m ∈ p.parts, Odd (p.parts.count m) → 2 * r + 1 ≤ p.parts.count m } =
    Set.ncard { p : Nat.Partition n |
      ∀ m ∈ p.parts, Odd m → m % (4 * r + 2) = 2 * r + 1 }

Sources

• Beaullah Mugwangwavari and Darlison Nyirenda, "A Note on the Andrews-Ericksson-Petrov-Romick Bijection for MacMahon's Partition Theorem", Journal of Integer Sequences 24 (2021), Article 21.5.6, Theorem andthm (Andrews' generalization), source lines 113–115, https://cs.uwaterloo.ca/journals/JIS/VOL24/Nyirenda/nyir26.tex. The paper attributes the result to Andrews.

TheoremMacMahon's partition theorem#✓fully proved

For every natural number \(n\), the number of partitions of \(n\) in which no part occurs exactly once equals the number of partitions of \(n\) all of whose parts are even or congruent to \(3\) modulo \(6\).

The source states the theorem for \(n \ge 1\); for \(n = 0\) both counts are \(1\), counting the empty partition.

theorem macmahon_multiplicity_one_free (n : ℕ) :
    (Finset.univ.filter (fun p : Nat.Partition n =>
      ∀ m ∈ p.parts, Multiset.count m p.parts ≠ 1)).card =
    (Nat.Partition.restricted n (fun m => m % 2 = 0 ∨ m % 6 = 3)).card

Sources

• Beaullah Mugwangwavari and Darlison Nyirenda, "A Note on the Andrews-Ericksson-Petrov-Romick Bijection for MacMahon's Partition Theorem", Journal of Integer Sequences 24 (2021), Article 21.5.6, Theorem MacMahon, source lines 88–97 with the displayed statement at lines 91–93, https://cs.uwaterloo.ca/journals/JIS/VOL24/Nyirenda/nyir26.tex. The paper attributes the theorem to P. A. MacMahon, Combinatory Analysis, Vol. 2 (1916).

2 fully proved

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