ℓ-regular cubic partitions with overlined odd parts

DefinitionGenerating function of ℓ-regular cubic partitions with odd parts overlined#✓fully proved

For a natural number \(\ell\), this is the integer power series

\[\frac{f_2 f_\ell^2 f_{4\ell}}{f_1^2 f_4 f_{2\ell}}, \qquad f_k = \prod_{m \ge 1} (1 - q^{km}).\]

It is meaningful for \(\ell \ge 1\), where the denominator has constant term \(1\) and so is invertible. For \(\ell > 2\), the range the paper considers, its coefficient of \(q^n\) is the number of \(\ell\)-regular cubic partitions of \(n\) with odd parts overlined.

public noncomputable def regularCubicOverlinedSeries (l : ℕ) : ℤ⟦X⟧ :=
  𝔣 2 * 𝔣 l ^ 2 * 𝔣 (4 * l) * invOfUnit (𝔣 1 ^ 2 * 𝔣 4 * 𝔣 (2 * l)) 1
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Section 1, eq. (2).
Uses 1 · Used by 2
Statement usesThe Euler products of partition theory
Used byℓ-regular cubic partitions with odd parts overlined · Generating function for \(\ell\)-regular cubic partitions with odd parts overlined

Definitionℓ-regular cubic partitions with odd parts overlined#✓fully proved

For natural numbers \(\ell\) and \(n\), \(a_\ell(n)\) is the coefficient of \(q^n\) in the integer power series

\[\frac{f_2 f_\ell^2 f_{4\ell}}{f_1^2 f_4 f_{2\ell}}, \qquad f_k = \prod_{m \ge 1} (1 - q^{km}).\]

The quotient is meaningful for \(\ell \ge 1\), where the denominator has constant term \(1\). For \(\ell > 2\), the range the paper considers, \(a_\ell(n)\) is the number of \(\ell\)-regular cubic partitions of \(n\) in which the first occurrence of each distinct odd part may be overlined; for example, \(a_3(5) = 18\).

public noncomputable def regularCubicOverlinedCount (l n : ℕ) : ℤ :=
  coeff n (regularCubicOverlinedSeries l)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Section 1, eq. (2).
Uses 1 · Used by 16
Statement usesGenerating function of ℓ-regular cubic partitions with odd parts overlined
Used byCongruences modulo 4 and 8 for 8-regular cubic partitions with odd parts overlined · Congruences modulo 16 for 8-regular cubic partitions with odd parts overlined · Congruences modulo 16 and 8 for 5-regular cubic partitions with odd parts overlined · Congruences modulo 8 for 5-regular cubic partitions with odd parts overlined · Congruences modulo 4 for 5-regular cubic partitions with odd parts overlined · Congruences modulo 8, 56 and 24 for 4-regular cubic partitions with odd parts overlined · Congruences modulo 4 for 4-regular cubic partitions with odd parts overlined · Generating function for \(\ell\)-regular cubic partitions with odd parts overlined · Congruences modulo 4 for regular cubic partitions with odd parts overlined · Congruence modulo 8 for 9-regular cubic partitions with odd parts overlined · Congruences modulo 4 for 9-regular cubic partitions with odd parts overlined · Congruences modulo 16 for 16-regular cubic partitions with odd parts overlined · Congruences modulo 18 and 12 for 3-regular cubic partitions with odd parts overlined · Congruences modulo 12 for 3-regular cubic partitions with odd parts overlined · Congruences modulo 6 for 3-regular cubic partitions with odd parts overlined · Constant term of the overlined ℓ-regular cubic partition numbers

TheoremGenerating function for \(\ell\)-regular cubic partitions with odd parts overlined#✓fully proved

Let \(\ell \ge 1\), write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\), and let \(a_\ell(n)\) be the \(n\)th coefficient of \(f_2 f_\ell^2 f_{4\ell} / (f_1^2 f_4 f_{2\ell})\) (the denominator has constant term \(1\), so it is invertible). Then

\[ f_1^2 f_4 f_{2\ell} \sum_{n \ge 0} a_\ell(n) q^n = f_2 f_\ell^2 f_{4\ell}. \]

For \(\ell > 2\) the paper identifies \(a_\ell(n)\) with the number of \(\ell\)-regular cubic partitions of \(n\) with odd parts overlined.

public theorem regularCubicOverlinedCount_generatingFunction {l : ℕ} (hl : 1 ≤ l) :
    𝔣 1 ^ 2 * 𝔣 4 * 𝔣 (2 * l) * PowerSeries.mk (regularCubicOverlinedCount l) =
      𝔣 2 * 𝔣 l ^ 2 * 𝔣 (4 * l)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Section 1, eq. (2).
Uses 4 · Used by 1
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined
Proof usesConstant term of the Euler products · Generating function of ℓ-regular cubic partitions with odd parts overlined
Used byConstant term of the overlined ℓ-regular cubic partition numbers

TheoremConstant term of the overlined ℓ-regular cubic partition numbers#✓fully proved

For every natural number \(\ell \ge 1\), the coefficient \(a_\ell(0)\) of \(q^0\) in the generating function of \(\ell\)-regular cubic partitions with odd parts overlined equals \(1\).

public theorem regularCubicOverlinedCount_zero {l : ℕ} (hl : 1 ≤ l) :
    regularCubicOverlinedCount l 0 = 1
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Section 1, eq. (2).
Uses 4
Statement usesℓ-regular cubic partitions with odd parts overlined
Proof usesConstant term of the Euler products · The Euler products of partition theory · Generating function for \(\ell\)-regular cubic partitions with odd parts overlined

TheoremFour 2-dissections of quotients of Euler products#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\). Then

\[ f_3 f_2^2 f_8 f_{12} f_{16} f_{24} f_{48} = f_1 \left(f_4 f_6 f_{16}^2 f_{24}^3 + q f_6 f_8^3 f_{12} f_{48}^2\right), \]
\[ f_1^3 f_4 f_6^2 f_{12} = f_3 \left(f_4^4 f_6^2 - 3 q f_2^2 f_{12}^4\right), \]
\[ f_5 f_2^3 f_8 f_{20} f_{40} = f_1 \left(f_2 f_8^2 f_{20}^3 + q f_4^3 f_{10} f_{40}^2\right), \]
\[ f_1 f_4 f_8 f_{10}^3 f_{40} = f_5 \left(f_2 f_8^2 f_{20}^3 - q f_4^3 f_{10} f_{40}^2\right). \]

These are the 2-dissections of \(f_3 / f_1\), \(f_1^3 / f_3\), \(f_5 / f_1\) and \(f_1 / f_5\) with the denominators cleared; for example, the first is \(f_3 / f_1 = \frac{f_4 f_6 f_{16} f_{24}^2}{f_2^2 f_8 f_{12} f_{48}} + q \frac{f_6 f_8^2 f_{48}}{f_2^2 f_{16} f_{24}}\) multiplied by \(f_1 f_2^2 f_8 f_{12} f_{16} f_{24} f_{48}\).

public theorem_wanted f3_f5_quotients_two_dissection :
    𝔣 3 * 𝔣 2 ^ 2 * 𝔣 8 * 𝔣 12 * 𝔣 16 * 𝔣 24 * 𝔣 48 =
        𝔣 1 * (𝔣 4 * 𝔣 6 * 𝔣 16 ^ 2 * 𝔣 24 ^ 3 + X * 𝔣 6 * 𝔣 8 ^ 3 * 𝔣 12 * 𝔣 48 ^ 2) ∧
      𝔣 1 ^ 3 * 𝔣 4 * 𝔣 6 ^ 2 * 𝔣 12 =
        𝔣 3 * (𝔣 4 ^ 4 * 𝔣 6 ^ 2 - 3 * X * 𝔣 2 ^ 2 * 𝔣 12 ^ 4) ∧
      𝔣 5 * 𝔣 2 ^ 3 * 𝔣 8 * 𝔣 20 * 𝔣 40 =
        𝔣 1 * (𝔣 2 * 𝔣 8 ^ 2 * 𝔣 20 ^ 3 + X * 𝔣 4 ^ 3 * 𝔣 10 * 𝔣 40 ^ 2) ∧
      𝔣 1 * 𝔣 4 * 𝔣 8 * 𝔣 10 ^ 3 * 𝔣 40 =
        𝔣 5 * (𝔣 2 * 𝔣 8 ^ 2 * 𝔣 20 ^ 3 - X * 𝔣 4 ^ 3 * 𝔣 10 * 𝔣 40 ^ 2)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Lemma 3, eqs. (9), (10), (13), (14).
Uses 1
Statement usesThe Euler products of partition theory

TheoremCongruences modulo 4 for regular cubic partitions with odd parts overlined#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\). For \(\ell \ge 1\) let \(a_\ell(n)\) be the \(n\)th coefficient of \(f_2 f_\ell^2 f_{4\ell} / (f_1^2 f_4 f_{2\ell})\); for \(\ell > 2\) this is the number of \(\ell\)-regular cubic partitions of \(n\) with odd parts overlined. Then for all natural numbers \(k\) and \(n\): (i) if \(k > 1\), then \(4\) divides \(a_{2k}(4n + 3)\); (ii) if \(k \ge 1\), then \(4\) divides \(a_{2k+1}(4n + 2)\); (iii) if \(k > 0\), then \(4\) divides \(a_{4k}(4n + 2)\); (iv) if \(k > 0\), then \(4\) divides \(a_{4k+1}(4n + 3)\).

public theorem_wanted regularCubicOverlinedCount_mod_four (k n : ℕ) :
    (1 < k → (4 : ℤ) ∣ regularCubicOverlinedCount (2 * k) (4 * n + 3)) ∧
      (1 ≤ k → (4 : ℤ) ∣ regularCubicOverlinedCount (2 * k + 1) (4 * n + 2)) ∧
      (0 < k → (4 : ℤ) ∣ regularCubicOverlinedCount (4 * k) (4 * n + 2)) ∧
      (0 < k → (4 : ℤ) ∣ regularCubicOverlinedCount (4 * k + 1) (4 * n + 3))
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 5, eqs. (17)–(20).
Uses 1
Statement usesℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 12 for 3-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_3(n)\) be the number of \(3\)-regular cubic partitions of \(n\) with odd parts overlined, and let \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-6}{p}\right) = -1\), and let \(\delta \ge 0\). Then for every \(n \ge 0\),

\[a_3\left(8p^{2\delta} n + \frac{10p^{2\delta} - 1}{3}\right) \equiv 6c_n \pmod{12},\]

where \(c_n\) is the coefficient of \(q^n\) in \(f_4 f_6\), and for all \(n \ge 0\) and all \(t\) with \(1 \le t \le p - 1\),

\[a_3\left(8p^{2\delta+2} n + 8p^{2\delta+1} t + \frac{10p^{2\delta+2} - 1}{3}\right) \equiv 0 \pmod{12}.\]

The divisions by \(3\) are exact.

public theorem_wanted regularCubicOverlinedCount_three_prime_mod_twelve (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-6) = -1) (δ : ℕ) :
    (∀ n : ℕ, regularCubicOverlinedCount 3 (8 * p ^ (2 * δ) * n + (10 * p ^ (2 * δ) - 1) / 3) ≡
        6 * coeff n (𝔣 4 * 𝔣 6) [ZMOD 12]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 3 (8 * p ^ (2 * δ + 2) * n + 8 * p ^ (2 * δ + 1) * t +
          (10 * p ^ (2 * δ + 2) - 1) / 3) ≡ 0 [ZMOD 12]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 6, eqs. (26), (27).
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 18 and 12 for 3-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_3(n)\) be the number of \(3\)-regular cubic partitions of \(n\) with odd parts overlined. For every \(n \ge 0\), \(a_3(8n + 5)\) is divisible by \(18\), and \(a_3(16n + 11)\) and \(a_3(16n + 15)\) are divisible by \(12\).

public theorem_wanted regularCubicOverlinedCount_three_mod_eighteen_twelve (n : ℕ) :
    (18 : ℤ) ∣ regularCubicOverlinedCount 3 (8 * n + 5) ∧
      (12 : ℤ) ∣ regularCubicOverlinedCount 3 (16 * n + 11) ∧
      (12 : ℤ) ∣ regularCubicOverlinedCount 3 (16 * n + 15)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Corollary 1, eqs. (39)–(41).
Uses 1
Statement usesℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 6 for 3-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_3(n)\) be the number of \(3\)-regular cubic partitions of \(n\) with odd parts overlined, and let \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-3}{p}\right) = -1\), and let \(\delta \ge 0\). Then for every \(n \ge 0\),

\[a_3\left(8p^{2\delta} n + \frac{4p^{2\delta} - 1}{3}\right) \equiv 2c_n \pmod 6,\]

where \(c_n\) is the coefficient of \(q^n\) in \(f_1 f_3\), and for all \(n \ge 0\) and all \(t\) with \(1 \le t \le p - 1\),

\[a_3\left(8p^{2\delta+2} n + 8p^{2\delta+1} t + \frac{4p^{2\delta+2} - 1}{3}\right) \equiv 0 \pmod 6.\]

The divisions by \(3\) are exact.

public theorem_wanted regularCubicOverlinedCount_three_prime_mod_six (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-3) = -1) (δ : ℕ) :
    (∀ n : ℕ, regularCubicOverlinedCount 3 (8 * p ^ (2 * δ) * n + (4 * p ^ (2 * δ) - 1) / 3) ≡
        2 * coeff n (𝔣 1 * 𝔣 3) [ZMOD 6]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 3 (8 * p ^ (2 * δ + 2) * n + 8 * p ^ (2 * δ + 1) * t +
          (4 * p ^ (2 * δ + 2) - 1) / 3) ≡ 0 [ZMOD 6]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 7, eq. (46) and the unnumbered congruence after it.
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 4 for 4-regular cubic partitions with odd parts overlined#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\), and \(a_4(n)\) for the \(n\)th coefficient of \(f_2 f_4^2 f_{16} / (f_1^2 f_4 f_8)\), the number of \(4\)-regular cubic partitions of \(n\) with odd parts overlined. Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-2}{p}\right) = -1\) and let \(\delta \ge 0\). Then for every \(n \ge 0\), \(a_4(4p^{2\delta} n + (3p^{2\delta} - 1)/2)\) is congruent modulo \(4\) to twice the coefficient of \(q^n\) in \(f_1 f_8\), and for all \(n \ge 0\) and \(1 \le t \le p - 1\),

\[ a_4\left(4p^{2\delta+2} n + 4p^{2\delta+1} t + \frac{3p^{2\delta+2} - 1}{2}\right) \equiv 0 \pmod{4}. \]

The divisions by \(2\) are exact.

public theorem_wanted regularCubicOverlinedCount_four_prime_mod_four (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-2) = -1) (δ : ℕ) :
    (∀ n : ℕ, regularCubicOverlinedCount 4 (4 * p ^ (2 * δ) * n + (3 * p ^ (2 * δ) - 1) / 2) ≡
        2 * coeff n (𝔣 1 * 𝔣 8) [ZMOD 4]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 4 (4 * p ^ (2 * δ + 2) * n + 4 * p ^ (2 * δ + 1) * t +
          (3 * p ^ (2 * δ + 2) - 1) / 2) ≡ 0 [ZMOD 4]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 8, eq. (49) and the unnumbered congruence after it.
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 8, 56 and 24 for 4-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_4(n)\) be the number of \(4\)-regular cubic partitions of \(n\) with odd parts overlined, that is, the \(n\)th coefficient of \(f_2 f_4^2 f_{16} / (f_1^2 f_4 f_8)\), where \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Then for every \(n \ge 0\), \(8\) divides \(a_4(4n + 3)\), \(56\) divides \(a_4(28n + 4j + 3)\) for each \(j = 1, \ldots, 6\), and \(24\) divides \(a_4(36n + 19)\).

public theorem_wanted regularCubicOverlinedCount_four_mod_eight_fiftySix_twentyFour (n : ℕ) :
    (8 : ℤ) ∣ regularCubicOverlinedCount 4 (4 * n + 3) ∧
      (∀ j ∈ Finset.Icc 1 6, (56 : ℤ) ∣ regularCubicOverlinedCount 4 (28 * n + 4 * j + 3)) ∧
      (24 : ℤ) ∣ regularCubicOverlinedCount 4 (36 * n + 19)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Corollary 2, eqs. (55)–(57).
Uses 1
Statement usesℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 8 for 5-regular cubic partitions with odd parts overlined#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\), and \(a_5(n)\) for the \(n\)th coefficient of \(f_2 f_5^2 f_{20} / (f_1^2 f_4 f_{10})\), the number of \(5\)-regular cubic partitions of \(n\) with odd parts overlined. Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-3}{p}\right) = -1\), let \(s = (p - 1)/6\) if \(p \equiv 1 \pmod 6\) and \(s = (-p - 1)/6\) otherwise, and let \(\delta \ge 0\). Then, comparing coefficients of power series in \(q\),

\[ \sum_{n \ge 0} a_5\left(10p^{2\delta} n + \frac{5p^{2\delta} - 2}{3}\right) q^n \equiv 2 (-1)^{\delta s} f_2^2 \pmod{8}, \]
\[ f_4 \sum_{n \ge 0} a_5\left(40p^{2\delta} n + \frac{65p^{2\delta} - 2}{3}\right) q^n \equiv 4 (-1)^{\delta s} f_1 f_8^2 \pmod{8}, \]

and for all \(n \ge 0\) and \(1 \le t \le p - 1\),

\[ a_5\left(10p^{2\delta+2} n + 10p^{2\delta+1} t + \frac{5p^{2\delta+2} - 2}{3}\right) \equiv 0 \pmod{8}, \]
\[ a_5\left(40p^{2\delta+2} n + 40p^{2\delta+1} t + \frac{65p^{2\delta+2} - 2}{3}\right) \equiv 0 \pmod{8}. \]

The divisions by \(3\) are exact. With \(\psi(q) = f_2^2 / f_1\) and \(\psi(q^4) = f_8^2 / f_4\), the power series congruences are the paper's \(\sum_n a_5(\ldots) q^n \equiv 2 (-1)^{\delta s} f_1 \psi(q) \pmod{8}\) and \(\sum_n a_5(\ldots) q^n \equiv 4 (-1)^{\delta s} f_1 \psi(q^4) \pmod{8}\), the second multiplied by \(f_4\).

public theorem_wanted regularCubicOverlinedCount_five_prime_mod_eight (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-3) = -1)
    (s : ℤ) (hs : s = if p % 6 = 1 then ((p : ℤ) - 1) / 6 else (-(p : ℤ) - 1) / 6) (δ : ℕ) :
    (∀ n : ℕ, regularCubicOverlinedCount 5 (10 * p ^ (2 * δ) * n + (5 * p ^ (2 * δ) - 2) / 3) ≡
        2 * (-1) ^ ((δ : ℤ) * s).natAbs * coeff n (𝔣 2 ^ 2) [ZMOD 8]) ∧
      (∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 5 (10 * p ^ (2 * δ + 2) * n + 10 * p ^ (2 * δ + 1) * t +
          (5 * p ^ (2 * δ + 2) - 2) / 3) ≡ 0 [ZMOD 8]) ∧
      (∀ n : ℕ, coeff n (𝔣 4 * PowerSeries.mk fun m =>
          regularCubicOverlinedCount 5 (40 * p ^ (2 * δ) * m + (65 * p ^ (2 * δ) - 2) / 3)) ≡
        4 * (-1) ^ ((δ : ℤ) * s).natAbs * coeff n (𝔣 1 * 𝔣 8 ^ 2) [ZMOD 8]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 5 (40 * p ^ (2 * δ + 2) * n + 40 * p ^ (2 * δ + 1) * t +
          (65 * p ^ (2 * δ + 2) - 2) / 3) ≡ 0 [ZMOD 8]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 9, eqs. (64)–(67).
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 16 and 8 for 5-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_5(n)\) be the number of \(5\)-regular cubic partitions of \(n\) with odd parts overlined, that is, the \(n\)th coefficient of \(f_2 f_5^2 f_{20} / (f_1^2 f_4 f_{10})\), where \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Then for every \(n \ge 0\), \(16\) divides \(a_5(8n + 7)\), and \(8\) divides \(a_5(20n + 2j + 1)\) for each \(j = 1, \ldots, 9\).

public theorem_wanted regularCubicOverlinedCount_five_mod_sixteen_eight (n : ℕ) :
    (16 : ℤ) ∣ regularCubicOverlinedCount 5 (8 * n + 7) ∧
      ∀ j ∈ Finset.Icc 1 9, (8 : ℤ) ∣ regularCubicOverlinedCount 5 (20 * n + 2 * j + 1)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Corollary 3, eqs. (76), (77).
Uses 1
Statement usesℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 4 for 5-regular cubic partitions with odd parts overlined#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\), and \(a_5(n)\) for the \(n\)th coefficient of \(f_2 f_5^2 f_{20} / (f_1^2 f_4 f_{10})\), the number of \(5\)-regular cubic partitions of \(n\) with odd parts overlined. Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-15}{p}\right) = -1\), let \(s = (p - 1)/6\) if \(p \equiv 1 \pmod 6\) and \(s = (-p - 1)/6\) otherwise, and let \(\delta \ge 0\). Then, comparing coefficients of power series in \(q\),

\[ f_{20} \sum_{n \ge 0} a_5\left(16p^{2\delta+2} n + \frac{122p^{2\delta+2} - 2}{3}\right) q^n \equiv 2 (-1)^{(\delta+1) s} f_1 f_{40}^2 \pmod{4}, \]

and for all \(n \ge 0\) and \(1 \le t \le p - 1\),

\[ a_5\left(16p^{2\delta+2} n + 16p^{2\delta+1} t + \frac{122p^{2\delta+2} - 2}{3}\right) \equiv 0 \pmod{4}. \]

The divisions by \(3\) are exact. With \(\psi(q^{20}) = f_{40}^2 / f_{20}\), the first congruence is the paper's \(\sum_n a_5(\ldots) q^n \equiv 2 (-1)^{(\delta+1) s} f_1 \psi(q^{20}) \pmod{4}\) multiplied by \(f_{20}\).

public theorem_wanted regularCubicOverlinedCount_five_prime_mod_four (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-15) = -1)
    (s : ℤ) (hs : s = if p % 6 = 1 then ((p : ℤ) - 1) / 6 else (-(p : ℤ) - 1) / 6) (δ : ℕ) :
    (∀ n : ℕ, coeff n (𝔣 20 * PowerSeries.mk fun m =>
          regularCubicOverlinedCount 5
            (16 * p ^ (2 * δ + 2) * m + (122 * p ^ (2 * δ + 2) - 2) / 3)) ≡
        2 * (-1) ^ (((δ : ℤ) + 1) * s).natAbs * coeff n (𝔣 1 * 𝔣 40 ^ 2) [ZMOD 4]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 5 (16 * p ^ (2 * δ + 2) * n + 16 * p ^ (2 * δ + 1) * t +
          (122 * p ^ (2 * δ + 2) - 2) / 3) ≡ 0 [ZMOD 4]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 10, eqs. (80), (81).
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 4 and 8 for 8-regular cubic partitions with odd parts overlined#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\), and \(a_8(n)\) for the \(n\)th coefficient of \(f_2 f_8^2 f_{32} / (f_1^2 f_4 f_{16})\), the number of \(8\)-regular cubic partitions of \(n\) with odd parts overlined. Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-3}{p}\right) = -1\), let \(s = (p - 1)/6\) if \(p \equiv 1 \pmod 6\) and \(s = (-p - 1)/6\) otherwise, and let \(\delta \ge 0\). Then, comparing coefficients of power series in \(q\),

\[ f_4 \sum_{n \ge 0} a_8\left(4p^{2\delta} n + \frac{13p^{2\delta} - 7}{6}\right) q^n \equiv 2 (-1)^{\delta s} f_1 f_8^2 \pmod{4}, \]
\[ f_1 \sum_{n \ge 0} a_8\left(4p^{2\delta} n + \frac{19p^{2\delta} - 7}{6}\right) q^n \equiv 4 (-1)^{\delta s} f_2^2 f_{16} \pmod{8}, \]

and for all \(n \ge 0\) and \(1 \le t \le p - 1\),

\[ a_8\left(4p^{2\delta+2} n + 4p^{2\delta+1} t + \frac{13p^{2\delta+2} - 7}{6}\right) \equiv 0 \pmod{4}, \]
\[ a_8\left(4p^{2\delta+2} n + 4p^{2\delta+1} t + \frac{19p^{2\delta+2} - 7}{6}\right) \equiv 0 \pmod{8}. \]

The divisions by \(6\) are exact. With \(\psi(q) = f_2^2 / f_1\) and \(\psi(q^4) = f_8^2 / f_4\), the power series congruences are the paper's \(\sum_n a_8(\ldots) q^n \equiv 2 (-1)^{\delta s} f_1 \psi(q^4) \pmod{4}\) and \(\sum_n a_8(\ldots) q^n \equiv 4 (-1)^{\delta s} \psi(q) f_{16} \pmod{8}\), multiplied by \(f_4\) and by \(f_1\).

MetaMathlibExt.regularCubicOverlinedCount_eight_prime_mod_four_eight · 15 lines 
public theorem_wanted regularCubicOverlinedCount_eight_prime_mod_four_eight (p : ℕ)
    [Fact p.Prime] (hp : 5 ≤ p) (hleg : legendreSym p (-3) = -1)
    (s : ℤ) (hs : s = if p % 6 = 1 then ((p : ℤ) - 1) / 6 else (-(p : ℤ) - 1) / 6) (δ : ℕ) :
    (∀ n : ℕ, coeff n (𝔣 4 * PowerSeries.mk fun m =>
          regularCubicOverlinedCount 8 (4 * p ^ (2 * δ) * m + (13 * p ^ (2 * δ) - 7) / 6)) ≡
        2 * (-1) ^ ((δ : ℤ) * s).natAbs * coeff n (𝔣 1 * 𝔣 8 ^ 2) [ZMOD 4]) ∧
      (∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 8 (4 * p ^ (2 * δ + 2) * n + 4 * p ^ (2 * δ + 1) * t +
          (13 * p ^ (2 * δ + 2) - 7) / 6) ≡ 0 [ZMOD 4]) ∧
      (∀ n : ℕ, coeff n (𝔣 1 * PowerSeries.mk fun m =>
          regularCubicOverlinedCount 8 (4 * p ^ (2 * δ) * m + (19 * p ^ (2 * δ) - 7) / 6)) ≡
        4 * (-1) ^ ((δ : ℤ) * s).natAbs * coeff n (𝔣 2 ^ 2 * 𝔣 16) [ZMOD 8]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 8 (4 * p ^ (2 * δ + 2) * n + 4 * p ^ (2 * δ + 1) * t +
          (19 * p ^ (2 * δ + 2) - 7) / 6) ≡ 0 [ZMOD 8]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 11, eqs. (93)–(96).
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 16 for 8-regular cubic partitions with odd parts overlined#●ready to prove

Write \(f_k = \prod_{m \ge 1} (1 - q^{km})\) for the Euler products in the ring of integer power series in \(q\), and \(a_8(n)\) for the \(n\)th coefficient of \(f_2 f_8^2 f_{32} / (f_1^2 f_4 f_{16})\), the number of \(8\)-regular cubic partitions of \(n\) with odd parts overlined. Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-6}{p}\right) = -1\), let \(s = (p - 1)/6\) if \(p \equiv 1 \pmod 6\) and \(s = (-p - 1)/6\) otherwise, and let \(\delta \ge 0\). Then, comparing coefficients of power series in \(q\),

\[ f_8 \sum_{n \ge 0} a_8\left(4p^{2\delta} n + \frac{25p^{2\delta} - 7}{6}\right) q^n \equiv 8 (-1)^{\delta s} f_1 f_{16}^2 \pmod{16}, \]

and for all \(n \ge 0\) and \(1 \le t \le p - 1\),

\[ a_8\left(4p^{2\delta+2} n + 4p^{2\delta+1} t + \frac{25p^{2\delta+2} - 7}{6}\right) \equiv 0 \pmod{16}. \]

The divisions by \(6\) are exact. With \(\psi(q^8) = f_{16}^2 / f_8\), the first congruence is the paper's \(\sum_n a_8(\ldots) q^n \equiv 8 (-1)^{\delta s} f_1 \psi(q^8) \pmod{16}\) multiplied by \(f_8\).

public theorem_wanted regularCubicOverlinedCount_eight_prime_mod_sixteen (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-6) = -1)
    (s : ℤ) (hs : s = if p % 6 = 1 then ((p : ℤ) - 1) / 6 else (-(p : ℤ) - 1) / 6) (δ : ℕ) :
    (∀ n : ℕ, coeff n (𝔣 8 * PowerSeries.mk fun m =>
          regularCubicOverlinedCount 8 (4 * p ^ (2 * δ) * m + (25 * p ^ (2 * δ) - 7) / 6)) ≡
        8 * (-1) ^ ((δ : ℤ) * s).natAbs * coeff n (𝔣 1 * 𝔣 16 ^ 2) [ZMOD 16]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 8 (4 * p ^ (2 * δ + 2) * n + 4 * p ^ (2 * δ + 1) * t +
          (25 * p ^ (2 * δ + 2) - 7) / 6) ≡ 0 [ZMOD 16]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 12, eqs. (105), (106).
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 4 for 9-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_9(n)\) be the number of \(9\)-regular cubic partitions of \(n\) with odd parts overlined, and let \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-2}{p}\right) = -1\), and let \(\delta \ge 0\). Then for every \(n \ge 0\),

\[a_9\left(8p^{2\delta} n + \frac{19p^{2\delta} - 4}{3}\right) \equiv 2c_n \pmod 4,\]

where \(c_n\) is the coefficient of \(q^n\) in \(f_1 f_{18}\), and for all \(n \ge 0\) and all \(t\) with \(1 \le t \le p - 1\),

\[a_9\left(8p^{2\delta+2} n + 8p^{2\delta+1} t + \frac{19p^{2\delta+2} - 4}{3}\right) \equiv 0 \pmod 4.\]

The divisions by \(3\) are exact.

public theorem_wanted regularCubicOverlinedCount_nine_prime_mod_four (p : ℕ) [Fact p.Prime]
    (hp : 5 ≤ p) (hleg : legendreSym p (-2) = -1) (δ : ℕ) :
    (∀ n : ℕ, regularCubicOverlinedCount 9 (8 * p ^ (2 * δ) * n + (19 * p ^ (2 * δ) - 4) / 3) ≡
        2 * coeff n (𝔣 1 * 𝔣 18) [ZMOD 4]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 9 (8 * p ^ (2 * δ + 2) * n + 8 * p ^ (2 * δ + 1) * t +
          (19 * p ^ (2 * δ + 2) - 4) / 3) ≡ 0 [ZMOD 4]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 13, eq. (110) and the unnumbered congruence after it.
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined

TheoremCongruence modulo 8 for 9-regular cubic partitions with odd parts overlined#●ready to prove

For every \(n \ge 0\), the number \(a_9(4n + 3)\) of \(9\)-regular cubic partitions of \(4n + 3\) with odd parts overlined is divisible by \(8\).

public theorem_wanted regularCubicOverlinedCount_nine_four_mul_add_three (n : ℕ) :
    (8 : ℤ) ∣ regularCubicOverlinedCount 9 (4 * n + 3)
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Corollary 4, eq. (118).
Uses 1
Statement usesℓ-regular cubic partitions with odd parts overlined

TheoremCongruences modulo 16 for 16-regular cubic partitions with odd parts overlined#●ready to prove

Let \(a_{16}(n)\) be the number of \(16\)-regular cubic partitions of \(n\) with odd parts overlined, and let \(f_k = \prod_{m \ge 1} (1 - q^{km})\). Let \(p \ge 5\) be a prime with Legendre symbol \(\left(\frac{-2}{p}\right) = -1\), and let \(\delta \ge 0\). Then for every \(n \ge 0\),

\[a_{16}\left(4p^{2\delta} n + \frac{11p^{2\delta} - 5}{2}\right) \equiv 8c_n \pmod{16},\]

where \(c_n\) is the coefficient of \(q^n\) in \(f_1 f_{32}\), and for all \(n \ge 0\) and all \(t\) with \(1 \le t \le p - 1\),

\[a_{16}\left(4p^{2\delta+2} n + 4p^{2\delta+1} t + \frac{11p^{2\delta+2} - 5}{2}\right) \equiv 0 \pmod{16}.\]

The divisions by \(2\) are exact.

public theorem_wanted regularCubicOverlinedCount_sixteen_prime_mod_sixteen (p : ℕ)
    [Fact p.Prime] (hp : 5 ≤ p) (hleg : legendreSym p (-2) = -1) (δ : ℕ) :
    (∀ n : ℕ, regularCubicOverlinedCount 16 (4 * p ^ (2 * δ) * n + (11 * p ^ (2 * δ) - 5) / 2) ≡
        8 * coeff n (𝔣 1 * 𝔣 32) [ZMOD 16]) ∧
      ∀ n t : ℕ, 1 ≤ t → t ≤ p - 1 →
        regularCubicOverlinedCount 16 (4 * p ^ (2 * δ + 2) * n + 4 * p ^ (2 * δ + 1) * t +
          (11 * p ^ (2 * δ + 2) - 5) / 2) ≡ 0 [ZMOD 16]
Sources
  • Pujashree Buragohain and Nipen Saikia, Some New Congruences for l-Regular Cubic Partitions with Odd Parts Overlined, INTEGERS 24 (2024), Article A101, Theorem 14, eq. (119) and the unnumbered congruence after it.
Uses 2
Statement usesThe Euler products of partition theory · ℓ-regular cubic partitions with odd parts overlined
4 fully proved15 ready to prove

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