Generalized cubic partitions

DefinitionGenerating function of generalized cubic partitions#✓fully proved

Let \(f_k = \prod_{m \ge 1} (1 - q^{km})\) be the Euler products. For a natural number \(c\), this is the integer power series \(1/(f_1 f_2^{c-1})\) in \(q\), the inverse of \(f_1 f_2^{c-1}\) (whose constant term is \(1\)). For \(c \ge 1\) its coefficient of \(q^n\) is the number of partitions of \(n\) whose even parts may each come in \(c\) colors. For \(c = 0\) the exponent \(c - 1\) is truncated to \(0\), so the series is then the inverse of \(f_1\).

public noncomputable def generalizedCubicPartitionSeries (c : ℕ) : ℤ⟦X⟧ :=
  invOfUnit (𝔣 1 * 𝔣 2 ^ (c - 1)) 1
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Section 1.
Uses 1 · Used by 2
Statement usesThe Euler products of partition theory
Used byGeneralized cubic partition numbers · Generating function identity for generalized cubic partition numbers

DefinitionGeneralized cubic partition numbers#✓fully proved

Let \(f_k = \prod_{m \ge 1} (1 - q^{km})\) be the Euler products. For natural numbers \(c\) and \(n\), \(a_c(n)\) is the integer coefficient of \(q^n\) in the generating function of generalized cubic partitions, the power series \(1/(f_1 f_2^{c-1})\). For \(c \ge 1\) it is the number of partitions of \(n\) whose even parts may each come in \(c\) colors; \(c = 1\) gives the partition numbers and \(c = 2\) Chan's cubic partition numbers. For \(c = 0\) the exponent \(c - 1\) is truncated to \(0\), so \(a_0(n)\) is also the number of partitions of \(n\).

public noncomputable def generalizedCubicPartitionCount (c n : ℕ) : ℤ :=
  coeff n (generalizedCubicPartitionSeries c)
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Section 1.
Uses 1 · Used by 5
Statement usesGenerating function of generalized cubic partitions
Used byGenerating function identity for generalized cubic partition numbers · Generalized cubic partition congruences for primes \(p \equiv 5, 7 \pmod 8\) · Generalized cubic partition congruences modulo primes \(p \ge 7\), \(p \equiv 3, 7 \pmod 8\) · Amdeberhan–Sellers–Singh congruences · Generalized cubic partitions of zero

TheoremGenerating function identity for generalized cubic partition numbers#✓fully proved

Let \(f_k = \prod_{m \ge 1} (1 - q^{km})\) be the Euler products and \(a_c(n)\) the generalized cubic partition numbers. For every natural number \(c\), \(f_1 f_2^{c-1} \sum_{n \ge 0} a_c(n) q^n = 1\) as integer power series in \(q\), where for \(c = 0\) the exponent \(c - 1\) is truncated to \(0\).

public theorem generalizedCubicPartitionCount_generatingFunction (c : ℕ) :
    𝔣 1 * 𝔣 2 ^ (c - 1) * PowerSeries.mk (generalizedCubicPartitionCount c) = 1
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Section 1.
Uses 4 · Used by 1
Statement usesGeneralized cubic partition numbers · The Euler products of partition theory
Proof usesConstant term of the Euler products · Generating function of generalized cubic partitions
Used byGeneralized cubic partitions of zero

TheoremGeneralized cubic partitions of zero#✓fully proved

For every natural number \(c\), the generalized cubic partition number \(a_c(0)\) equals \(1\).

public theorem generalizedCubicPartitionCount_zero (c : ℕ) :
    generalizedCubicPartitionCount c 0 = 1
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Section 1.
Uses 4
Statement usesGeneralized cubic partition numbers
Proof usesConstant term of the Euler products · Generating function identity for generalized cubic partition numbers · The Euler products of partition theory

TheoremAmdeberhan–Sellers–Singh congruences#●ready to prove

For every natural number \(n\), the generalized cubic partition numbers satisfy \(a_3(7n + 4) \equiv 0 \pmod{7}\) and \(a_5(11n + 10) \equiv 0 \pmod{11}\).

public theorem_wanted generalizedCubicPartitionCount_seven_eleven (n : ℕ) :
    (7 : ℤ) ∣ generalizedCubicPartitionCount 3 (7 * n + 4) ∧
      (11 : ℤ) ∣ generalizedCubicPartitionCount 5 (11 * n + 10)
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Theorem 1 (quoting Amdeberhan, Sellers, and Singh), Equations (1) and (2).
Uses 1
Statement usesGeneralized cubic partition numbers

TheoremGeneralized cubic partition congruences for primes \(p \equiv 5, 7 \pmod 8\)#●ready to prove

Let \(a_c(n)\) be the generalized cubic partition numbers, let \(p\) be a prime with \(p \equiv 5\) or \(p \equiv 7 \pmod 8\), and let \(l\) be a natural number with \(p \mid 8l + 3\). Then \(a_{p-4}(pn + l) \equiv 0 \pmod p\) for every natural number \(n\).

The source assumes \(0 \le l \le p - 1\); the statement for a larger \(l\) follows from the one for \(l \bmod p\), which satisfies the same divisibility.

public theorem_wanted generalizedCubicPartitionCount_prime_five_seven_mod_eight {p : ℕ}
    (hp : p.Prime) (hp8 : p % 8 = 5 ∨ p % 8 = 7) {l : ℕ} (hdvd : p ∣ 8 * l + 3) (n : ℕ) :
    (p : ℤ) ∣ generalizedCubicPartitionCount (p - 4) (p * n + l)
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Theorem 2, Equation (9).
Uses 1
Statement usesGeneralized cubic partition numbers

TheoremGeneralized cubic partition congruences modulo primes \(p \ge 7\), \(p \equiv 3, 7 \pmod 8\)#●ready to prove

Let \(a_c(n)\) be the generalized cubic partition numbers and let \(p \ge 7\) be a prime with \(p \equiv 3\) or \(p \equiv 7 \pmod 8\). Then \(a_{p-6}\bigl(pn + 13(p^2 - 1)/24\bigr) \equiv 0 \pmod p\) for every natural number \(n\).

Since \(24 \mid p^2 - 1\), the shift \(13(p^2 - 1)/24\) is a natural number.

public theorem_wanted generalizedCubicPartitionCount_prime_three_seven_mod_eight {p : ℕ}
    (hp : p.Prime) (hp7 : 7 ≤ p) (hp8 : p % 8 = 3 ∨ p % 8 = 7) (n : ℕ) :
    (p : ℤ) ∣ generalizedCubicPartitionCount (p - 6) (p * n + 13 * (p ^ 2 - 1) / 24)
Sources
  • Russelle Guadalupe, A Note on Congruences for Generalized Cubic Partitions Modulo Primes, INTEGERS 25 (2025), Article A20, Theorem 3, Equation (13).
Uses 1
Statement usesGeneralized cubic partition numbers
4 fully proved3 ready to prove

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