Young diagrams and partition statistics

Statistics and structure of a single partition: conjugation, hook lengths, s-cores, perimeter, diagonal sequences, lit cells, rank and crank, max-rook numbers, perfect and double-perfect partitions, and reconstruction of a partition from pairwise products.

• Balanceable partitions
• Compositions, conjugate partitions and dominance order
• Diagonal sequences of integer partitions
• Fixed hooks in a column of a partition
• Lit cells in integer partitions
• Max-rook numbers of partitions
• Number of values of diversity indices
• Perfect and double-perfect partitions
• Reconstructing partitions from pairwise products
• Simultaneous core partitions

TheoremPartitions with non-negative crank#✓fully proved

For a partition \(\lambda\), let \(\omega(\lambda)\) be the number of parts equal to \(1\). The crank of \(\lambda\) is its largest part if \(\omega(\lambda) = 0\) (the largest part of the empty partition being \(0\)), and otherwise the number of parts greater than \(\omega(\lambda)\), counted with multiplicity, minus \(\omega(\lambda)\). Let \(p(k)\) be the number of partitions of \(k\). Then for every natural number \(N\), the number of partitions of \(N\) with non-negative crank equals

\[\sum_{n \ge 0,\ n(n+1)/2 \le N} (-1)^n \, p\bigl(N - n(n+1)/2\bigr).\]

This is the coefficient-wise form of the generating function for partitions with non-negative crank. The source notes this result is originally due to Uncu.

public theorem crank_nonneg_partition_count
    (N : ℕ) :
    Fintype.card { P : Nat.Partition N //
      0 ≤ (if Multiset.count 1 P.parts = 0 then (P.parts.sup : ℤ)
        else (((P.parts.filter
          (fun i => Multiset.count 1 P.parts < i)).card : ℤ)
          - (Multiset.count 1 P.parts : ℤ)))}
    = ∑ n ∈ Finset.range (N + 1),
        (if (n + 1) * n / 2 ≤ N
          then (-1 : ℤ) ^ n * (Fintype.card (Nat.Partition (N - (n + 1) * n / 2)) : ℤ)
          else 0)

Sources

• George E. Andrews and David Newman, "The Minimal Excludant in Integer Partitions," Journal of Integer Sequences 23 (2020), Article 20.2.3, Theorem (equation 1.11), lines 161–165, https://cs.uwaterloo.ca/journals/JIS/VOL23/Andrews/andrews5.tex

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