Merca's identities for sums of different parts

DefinitionSum of different parts congruent to \(r\) modulo \(m\)#✓fully proved

For natural numbers \(m\), \(r\), \(n\), \(a_{r,m}(n)\) is the sum, over all partitions \(\mu\) of \(n\), of the different parts \(x\) of \(\mu\) with \(x \bmod m = r\); a part contributes \(x\) once regardless of its multiplicity. It is used with \(0 \le r < m\), where \(x \bmod m = r\) means \(x \equiv r \pmod m\).

public def sumDistinctPartsModEq (m r n : ℕ) : ℕ :=
  ∑ μ : n.Partition, ∑ x ∈ μ.parts.toFinset.filter (fun x => x % m = r), x
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 1 (definition of a_{r,m}(n)).
Used by 9
Used byDifferent parts in a residue class and Garden of Eden partitions · Merca's identity for different parts in a residue class · Different parts in a residue class and partitions of nonnegative rank · Different parts in a residue class and partitions of positive rank · Different parts in a residue class and even-minus-odd partition counts · Signed triangular sum of \(a_{r,m}\) and distinct partitions · Different parts in a residue class and partitions of nonnegative crank · Twice-square sum of \(a_{r,m}\) and distinct odd parts · Small values of Merca's distinct-part sums

DefinitionSum of different parts appearing at least \(m\) times#✓fully proved

For natural numbers \(m\) and \(n\), \(s_m(n)\) is the sum, over all partitions \(\mu\) of \(n\), of the different parts of \(\mu\) that appear at least \(m\) times in \(\mu\); such a part \(j\) contributes \(j\) once regardless of its multiplicity.

public def sumDistinctPartsMultAtLeast (m n : ℕ) : ℕ :=
  ∑ μ : n.Partition, ∑ x ∈ μ.parts.toFinset.filter (fun x => m ≤ μ.parts.count x), x
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 1 (definition of s_m(n)).
Used by 7
Used byGarden of Eden identity for \(s_m\) · Merca's identity for \(s_m(n)\) · Pentagonal sum of \(s_m\) and nonnegative rank · Pentagonal sum of \(s_m\) and positive rank · Signed triangular sum of \(s_m\) and distinct partitions · Signed triangular sum of \(s_m\) and nonnegative crank · Small values of Merca's distinct-part sums

TheoremSmall values of Merca's distinct-part sums#✓fully proved

For natural numbers \(m\), \(r\) and \(n\), let \(a_{r,m}(n)\) be the sum, over all partitions of \(n\), of the different parts \(x\) of the partition with \(x \bmod m = r\), and let \(s_m(n)\) be the sum, over all partitions of \(n\), of the different parts that occur at least \(m\) times in the partition; each such part counts once per partition. Then \(a_{r,m}(0) = 0\) and \(s_m(0) = 0\) for all \(m\) and \(r\), and moreover \(a_{0,1}(1) = 1\), \(s_1(1) = 1\) and \(s_2(1) = 0\).

The empty partition of \(0\) has no parts, and \(1\) has the single partition \(1\).

public theorem sumDistinctParts_small_values (m r : ℕ) :
    sumDistinctPartsModEq m r 0 = 0 ∧ sumDistinctPartsMultAtLeast m 0 = 0 ∧
      sumDistinctPartsModEq 1 0 1 = 1 ∧ sumDistinctPartsMultAtLeast 1 1 = 1 ∧
      sumDistinctPartsMultAtLeast 2 1 = 0
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 1 (definitions of a_{r,m}(n) and s_m(n)).
Uses 2
Statement usesSum of different parts congruent to \(r\) modulo \(m\) · Sum of different parts appearing at least \(m\) times

DefinitionDyson's rank of a partition#✓fully proved

The rank of a partition \(\mu\) of \(n\) is the integer \(r(\mu) = \mu_1 - \ell(\mu)\), the largest part \(\mu_1\) of \(\mu\) minus its number of parts \(\ell(\mu)\), where the largest part of the empty partition is taken to be \(0\). In particular the empty partition has rank \(0\).

public def dysonRank {n : ℕ} (μ : n.Partition) : ℤ :=
  (μ.parts.sup : ℤ) - μ.parts.card
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 2 (definition of the rank before Theorem 8).
Used by 7
Used byRank of the partitions of \(0\) and \(1\) · Different parts in a residue class and Garden of Eden partitions · Different parts in a residue class and partitions of nonnegative rank · Different parts in a residue class and partitions of positive rank · Garden of Eden identity for \(s_m\) · Pentagonal sum of \(s_m\) and nonnegative rank · Pentagonal sum of \(s_m\) and positive rank

TheoremRank of the partitions of \(0\) and \(1\)#✓fully proved

The empty partition of \(0\) and the partition \((1)\) of \(1\) both have Dyson rank \(0\).

Consequently the number of partitions of \(k\) with nonnegative rank is \(1\) for \(k = 0\) and \(k = 1\), no partition of \(1\) has positive rank, and no partition of \(0\) has rank at most \(-2\).

public theorem dysonRank_small_values (μ : Nat.Partition 0) (ν : Nat.Partition 1) :
    dysonRank μ = 0 ∧ dysonRank ν = 0
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 2 (definition of the rank before Theorem 8).
Uses 1
Statement usesDyson's rank of a partition

DefinitionCrank of a partition#✓fully proved

Let \(\mu\) be a partition of \(n\) and let \(\omega(\mu)\) be the number of parts of \(\mu\) equal to \(1\). The crank of \(\mu\) (Andrews and Garvan) is the integer defined as follows: if \(\omega(\mu) = 0\), it is the largest part of \(\mu\), where the largest part of the empty partition is taken to be \(0\); if \(\omega(\mu) > 0\), it is the number of parts of \(\mu\) larger than \(\omega(\mu)\), counted with multiplicity, minus \(\omega(\mu)\).

public def partitionCrank {n : ℕ} (μ : n.Partition) : ℤ :=
  if μ.parts.count 1 = 0 then (μ.parts.sup : ℤ)
  else ((μ.parts.filter (fun x => μ.parts.count 1 < x)).card : ℤ) - μ.parts.count 1
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 2 (definition of the crank before Theorem 11).
Used by 3
Used byCrank of the partitions of \(0\) and \(1\) · Different parts in a residue class and partitions of nonnegative crank · Signed triangular sum of \(s_m\) and nonnegative crank

TheoremCrank of the partitions of \(0\) and \(1\)#✓fully proved

The empty partition of \(0\) has crank \(0\), and the partition \((1)\) of \(1\) has crank \(0 - 1 = -1\).

Consequently the number of partitions of \(0\) with nonnegative crank is \(1\), and no partition of \(1\) has nonnegative crank.

public theorem partitionCrank_small_values (μ : Nat.Partition 0) (ν : Nat.Partition 1) :
    partitionCrank μ = 0 ∧ partitionCrank ν = -1
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Section 2 (definition of the crank before Theorem 11).
Uses 1
Statement usesCrank of a partition

TheoremMerca's identity for different parts in a residue class#●ready to prove

Let \(m\), \(r\) and \(n\) be natural numbers with \(r < m\), let \(a_{r,m}(n)\) be the sum, over all partitions of \(n\), of the different parts congruent to \(r\) modulo \(m\), each counted once per partition whatever its multiplicity, and let \(p(k)\) be the number of partitions of \(k\). Then

\[a_{r,m}(n) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, p(n - mj - r).\]
public theorem_wanted sumDistinctPartsModEq_eq_sum_partition (m r n : ℕ) (hrm : r < m) :
    sumDistinctPartsModEq m r n =
      ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
        (m * j + r) * Fintype.card (Nat.Partition (n - (m * j + r)))
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 1 (i).
Uses 1
Statement usesSum of different parts congruent to \(r\) modulo \(m\)

TheoremMerca's identity for \(s_m(n)\)#●ready to prove

Let \(m \ge 1\) and \(n\) be natural numbers, write \(s_m(n)\) for the sum of the different parts that appear at least \(m\) times in partitions of \(n\), and \(p(k)\) for the number of partitions of \(k\). Then \(s_m(n) = \sum_{j \ge 0,\ mj \le n} j\, p(n - mj)\).

The paper's hypothesis \(0 \le r < m\) forces \(m \ge 1\); for \(m = 0\) the right side diverges.

public theorem_wanted sumDistinctPartsMultAtLeast_eq_sum_partition (m n : ℕ) (hm : 0 < m) :
    sumDistinctPartsMultAtLeast m n =
      ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j ≤ n),
        j * Fintype.card (Nat.Partition (n - m * j))
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 1 (ii).
Uses 1
Statement usesSum of different parts appearing at least \(m\) times

TheoremDifferent parts in a residue class and even-minus-odd partition counts#●ready to prove

Let \(m\), \(r\) and \(n\) be natural numbers with \(r < m\), and for \(k \ge 0\) let \(a_{r,m}(k)\) be the sum, over all partitions of \(k\), of the different parts congruent to \(r\) modulo \(m\), each counted once per partition whatever its multiplicity. Let \(p_{e-o}(k) = \sum_{\mu} (-1)^{\ell(\mu)}\), the sum over the partitions \(\mu\) of \(k\) with \(\ell(\mu)\) the number of parts of \(\mu\); it is the number of partitions of \(k\) with an even number of parts minus the number with an odd number of parts. Then

\[a_{r,m}(n) + 2 \sum_{k \ge 1,\ k^2 \le n} (-1)^k a_{r,m}(n - k^2) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, p_{e-o}(n - mj - r).\]
public theorem_wanted sumDistinctPartsModEq_square_eq_evenOdd (m r n : ℕ) (hrm : r < m) :
    (sumDistinctPartsModEq m r n : ℤ)
        + 2 * ∑ k ∈ (Finset.Icc 1 n).filter (fun k => k ^ 2 ≤ n),
          (-1 : ℤ) ^ k * (sumDistinctPartsModEq m r (n - k ^ 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) *
            ∑ μ : Nat.Partition (n - (m * j + r)), (-1 : ℤ) ^ μ.parts.card
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 2.
Uses 1
Statement usesSum of different parts congruent to \(r\) modulo \(m\)

TheoremTwice-square sum of \(a_{r,m}\) and distinct odd parts#●ready to prove

Let \(m, r, n\) be natural numbers with \(r < m\). Write \(a_{r,m}(k)\) for the sum of the different parts congruent to \(r\) modulo \(m\) in all partitions of \(k\), and \(q_{\mathrm{odd}}(k)\) for the number of partitions of \(k\) into distinct odd parts. Then

\[ a_{r,m}(n) + 2 \sum_{k \ge 1,\ 2k^2 \le n} (-1)^k a_{r,m}(n - 2k^2) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, q_{\mathrm{odd}}(n - mj - r). \]
public theorem_wanted sumDistinctPartsModEq_twiceSquare_eq_oddDistinct (m r n : ℕ)
    (hrm : r < m) :
    (sumDistinctPartsModEq m r n : ℤ)
        + 2 * ∑ k ∈ (Finset.Icc 1 n).filter (fun k => 2 * k ^ 2 ≤ n),
          (-1 : ℤ) ^ k * (sumDistinctPartsModEq m r (n - 2 * k ^ 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) * ((Nat.Partition.oddDistincts (n - (m * j + r))).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 4.
Uses 1
Statement usesSum of different parts congruent to \(r\) modulo \(m\)

TheoremSigned triangular sum of \(a_{r,m}\) and distinct partitions#●ready to prove

Let \(m, r, n\) be natural numbers with \(r < m\). Write \(a_{r,m}(k)\) for the sum of the different parts congruent to \(r\) modulo \(m\) in all partitions of \(k\), \(q(k)\) for the number of partitions of \(k\) into distinct parts, and \(\tilde q(\ell)\) for \(q(\ell/2)\) when \(\ell\) is even and for \(0\) when \(\ell\) is odd. Then

\[ \sum_{k \ge 0,\ k(k+1)/2 \le n} (-1)^{k(k+1)/2} a_{r,m}\bigl(n - k(k+1)/2\bigr) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, \tilde q(n - mj - r). \]
public theorem_wanted sumDistinctPartsModEq_triangular_eq_distinct_half (m r n : ℕ)
    (hrm : r < m) :
    ∑ k ∈ (Finset.range (n + 1)).filter (fun k => k * (k + 1) / 2 ≤ n),
        (-1 : ℤ) ^ (k * (k + 1) / 2) * (sumDistinctPartsModEq m r (n - k * (k + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) *
            if Even (n - (m * j + r)) then
              ((Nat.Partition.distincts ((n - (m * j + r)) / 2)).card : ℤ)
            else 0
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 6.
Uses 1
Statement usesSum of different parts congruent to \(r\) modulo \(m\)

TheoremSigned triangular sum of \(s_m\) and distinct partitions#●ready to prove

Let \(m \ge 1\) and \(n\) be natural numbers. Write \(s_m(k)\) for the sum of the different parts that appear at least \(m\) times in partitions of \(k\), \(q(k)\) for the number of partitions of \(k\) into distinct parts, and \(\tilde q(\ell)\) for \(q(\ell/2)\) when \(\ell\) is even and for \(0\) when \(\ell\) is odd. Then

\[ \sum_{k \ge 0,\ k(k+1)/2 \le n} (-1)^{k(k+1)/2} s_m\bigl(n - k(k+1)/2\bigr) = \sum_{j \ge 0,\ mj \le n} j\, \tilde q(n - mj). \]

The paper says \(m \ge 0\); for \(m = 0\) the right side diverges.

public theorem_wanted sumDistinctPartsMultAtLeast_triangular_eq_distinct_half (m n : ℕ)
    (hm : 0 < m) :
    ∑ k ∈ (Finset.range (n + 1)).filter (fun k => k * (k + 1) / 2 ≤ n),
        (-1 : ℤ) ^ (k * (k + 1) / 2) * (sumDistinctPartsMultAtLeast m (n - k * (k + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j ≤ n),
          (j : ℤ) *
            if Even (n - m * j) then ((Nat.Partition.distincts ((n - m * j) / 2)).card : ℤ)
            else 0
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 7.
Uses 1
Statement usesSum of different parts appearing at least \(m\) times

TheoremDifferent parts in a residue class and partitions of nonnegative rank#●ready to prove

Let \(m\), \(r\) and \(n\) be natural numbers with \(r < m\), and for \(k \ge 0\) let \(a_{r,m}(k)\) be the sum, over all partitions of \(k\), of the different parts congruent to \(r\) modulo \(m\), each counted once per partition whatever its multiplicity. Let \(N(k)\) be the number of partitions of \(k\) whose Dyson rank (largest part minus number of parts) is nonnegative. Then

\[\sum_{j \ge 0,\ j(3j+1)/2 \le n} (-1)^j a_{r,m}(n - j(3j+1)/2) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, N(n - mj - r).\]
public theorem_wanted sumDistinctPartsModEq_pentagonal_eq_nonnegRank (m r n : ℕ)
    (hrm : r < m) :
    ∑ j ∈ (Finset.range (n + 1)).filter (fun j => j * (3 * j + 1) / 2 ≤ n),
        (-1 : ℤ) ^ j * (sumDistinctPartsModEq m r (n - j * (3 * j + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - (m * j + r)) =>
              0 ≤ dysonRank μ).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 8 (i).
Uses 2
Statement usesDyson's rank of a partition · Sum of different parts congruent to \(r\) modulo \(m\)

TheoremDifferent parts in a residue class and partitions of positive rank#●ready to prove

Let \(m\), \(r\) and \(n\) be natural numbers with \(r < m\), and for \(k \ge 0\) let \(a_{r,m}(k)\) be the sum, over all partitions of \(k\), of the different parts congruent to \(r\) modulo \(m\), each counted once per partition whatever its multiplicity. Let \(R(k)\) be the number of partitions of \(k\) whose Dyson rank (largest part minus number of parts) is positive. Then

\[\sum_{j \ge 1,\ j(3j+1)/2 \le n} (-1)^{j+1} a_{r,m}(n - j(3j+1)/2) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, R(n - mj - r).\]
public theorem_wanted sumDistinctPartsModEq_pentagonal_eq_posRank (m r n : ℕ) (hrm : r < m) :
    ∑ j ∈ (Finset.Icc 1 n).filter (fun j => j * (3 * j + 1) / 2 ≤ n),
        (-1 : ℤ) ^ (j + 1) * (sumDistinctPartsModEq m r (n - j * (3 * j + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - (m * j + r)) =>
              0 < dysonRank μ).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 8 (ii).
Uses 2
Statement usesDyson's rank of a partition · Sum of different parts congruent to \(r\) modulo \(m\)

TheoremPentagonal sum of \(s_m\) and nonnegative rank#●ready to prove

Let \(m \ge 1\) and \(n\) be natural numbers. Write \(s_m(k)\) for the sum of the different parts that appear at least \(m\) times in partitions of \(k\), and \(N(k)\) for the number of partitions of \(k\) with nonnegative Dyson rank. Then

\[ \sum_{j \ge 0,\ j(3j+1)/2 \le n} (-1)^j s_m\bigl(n - j(3j+1)/2\bigr) = \sum_{j \ge 0,\ mj \le n} j\, N(n - mj). \]

The paper says \(m \ge 0\); for \(m = 0\) the right side diverges.

public theorem_wanted sumDistinctPartsMultAtLeast_pentagonal_eq_nonnegRank (m n : ℕ)
    (hm : 0 < m) :
    ∑ j ∈ (Finset.range (n + 1)).filter (fun j => j * (3 * j + 1) / 2 ≤ n),
        (-1 : ℤ) ^ j * (sumDistinctPartsMultAtLeast m (n - j * (3 * j + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j ≤ n),
          (j : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - m * j) => 0 ≤ dysonRank μ).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 9 (i).
Uses 2
Statement usesDyson's rank of a partition · Sum of different parts appearing at least \(m\) times

TheoremPentagonal sum of \(s_m\) and positive rank#●ready to prove

Let \(m \ge 1\) and \(n\) be natural numbers. Write \(s_m(k)\) for the sum of the different parts that appear at least \(m\) times in partitions of \(k\), and \(R(k)\) for the number of partitions of \(k\) with positive Dyson rank. Then

\[ \sum_{j \ge 1,\ j(3j+1)/2 \le n} (-1)^{j+1} s_m\bigl(n - j(3j+1)/2\bigr) = \sum_{j \ge 0,\ mj \le n} j\, R(n - mj). \]

The paper says \(m \ge 0\); for \(m = 0\) the right side diverges.

public theorem_wanted sumDistinctPartsMultAtLeast_pentagonal_eq_posRank (m n : ℕ)
    (hm : 0 < m) :
    ∑ j ∈ (Finset.Icc 1 n).filter (fun j => j * (3 * j + 1) / 2 ≤ n),
        (-1 : ℤ) ^ (j + 1) * (sumDistinctPartsMultAtLeast m (n - j * (3 * j + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j ≤ n),
          (j : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - m * j) => 0 < dysonRank μ).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 9 (ii).
Uses 2
Statement usesDyson's rank of a partition · Sum of different parts appearing at least \(m\) times

TheoremDifferent parts in a residue class and Garden of Eden partitions#●ready to prove

Let \(m\), \(r\) and \(n\) be natural numbers with \(r < m\), and for \(k \ge 0\) let \(a_{r,m}(k)\) be the sum, over all partitions of \(k\), of the different parts congruent to \(r\) modulo \(m\), each counted once per partition whatever its multiplicity. Let \(G(k)\) be the number of Garden of Eden partitions of \(k\), that is, partitions of \(k\) whose Dyson rank (largest part minus number of parts) is at most \(-2\). Then

\[\sum_{j \ge 1,\ 3j(j+1)/2 \le n} (-1)^{j+1} a_{r,m}(n - 3j(j+1)/2) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, G(n - mj - r).\]

The sum on the left starts at \(j = 1\). Starting it at \(j = 0\) makes the identity false: for \(m = 1\), \(r = 0\) and \(n = 1\) the left side becomes \(-a_{0,1}(1) = -1\) while the right side is \(0\).

public theorem_wanted sumDistinctPartsModEq_eq_gardenOfEden (m r n : ℕ) (hrm : r < m) :
    ∑ j ∈ (Finset.Icc 1 n).filter (fun j => 3 * (j * (j + 1)) / 2 ≤ n),
        (-1 : ℤ) ^ (j + 1) * (sumDistinctPartsModEq m r (n - 3 * (j * (j + 1)) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - (m * j + r)) =>
              dysonRank μ ≤ -2).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 10 (i).
Uses 2
Statement usesDyson's rank of a partition · Sum of different parts congruent to \(r\) modulo \(m\)

TheoremGarden of Eden identity for \(s_m\)#●ready to prove

Let \(m \ge 1\) and \(n\) be natural numbers. Write \(s_m(k)\) for the sum of the different parts that appear at least \(m\) times in partitions of \(k\), and \(G(k)\) for the number of Garden of Eden partitions of \(k\), those whose Dyson rank is at most \(-2\) (so \(G(0) = 0\)). Then

\[ \sum_{j \ge 1,\ 3j(j+1)/2 \le n} (-1)^{j+1} s_m\bigl(n - 3j(j+1)/2\bigr) = \sum_{j \ge 0,\ mj \le n} j\, G(n - mj). \]

The paper says \(m \ge 0\); for \(m = 0\) the right side diverges.

public theorem_wanted sumDistinctPartsMultAtLeast_eq_gardenOfEden (m n : ℕ) (hm : 0 < m) :
    ∑ j ∈ (Finset.Icc 1 n).filter (fun j => 3 * (j * (j + 1)) / 2 ≤ n),
        (-1 : ℤ) ^ (j + 1) * (sumDistinctPartsMultAtLeast m (n - 3 * (j * (j + 1)) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j ≤ n),
          (j : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - m * j) =>
              dysonRank μ ≤ -2).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 10 (ii).
Uses 2
Statement usesDyson's rank of a partition · Sum of different parts appearing at least \(m\) times

TheoremDifferent parts in a residue class and partitions of nonnegative crank#●ready to prove

Let \(m\), \(r\) and \(n\) be natural numbers with \(r < m\), and for \(k \ge 0\) let \(a_{r,m}(k)\) be the sum, over all partitions of \(k\), of the different parts congruent to \(r\) modulo \(m\), each counted once per partition whatever its multiplicity. Let \(C(k)\) be the number of partitions of \(k\) whose crank is nonnegative. Then

\[\sum_{j \ge 0,\ j(j+1)/2 \le n} (-1)^j a_{r,m}(n - j(j+1)/2) = \sum_{j \ge 0,\ mj + r \le n} (mj + r)\, C(n - mj - r).\]

The companion identity of the source with the number \(D(k)\) of partitions of \(k\) with positive crank is not stated: it fails for this crank, since \(D(1) = 0\) while \(p(1) - C(1) = 1\), where \(p(k)\) is the number of partitions of \(k\) (for \(m = 2\), \(r = 1\), \(n = 2\) its left side is \(1\) and its right side is \(0\)).

public theorem_wanted sumDistinctPartsModEq_triangular_eq_nonnegCrank (m r n : ℕ)
    (hrm : r < m) :
    ∑ j ∈ (Finset.range (n + 1)).filter (fun j => j * (j + 1) / 2 ≤ n),
        (-1 : ℤ) ^ j * (sumDistinctPartsModEq m r (n - j * (j + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j + r ≤ n),
          ((m * j + r : ℕ) : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - (m * j + r)) =>
              0 ≤ partitionCrank μ).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 11 (i).
Uses 2
Statement usesCrank of a partition · Sum of different parts congruent to \(r\) modulo \(m\)

TheoremSigned triangular sum of \(s_m\) and nonnegative crank#●ready to prove

Let \(m \ge 1\) and \(n\) be natural numbers. Write \(s_m(k)\) for the sum of the different parts that appear at least \(m\) times in partitions of \(k\), and \(C(k)\) for the number of partitions of \(k\) with nonnegative crank. Then

\[ \sum_{j \ge 0,\ j(j+1)/2 \le n} (-1)^j s_m\bigl(n - j(j+1)/2\bigr) = \sum_{j \ge 0,\ mj \le n} j\, C(n - mj). \]

The paper says \(m \ge 0\); for \(m = 0\) the right side diverges. Part (ii) (with \(D(k)\), positive crank) is omitted: it is false for this crank (\(m = 1\), \(n = 2\): left side \(s_1(1) = 1\), right side \(D(1) + 2 D(0) = 0\)).

public theorem_wanted sumDistinctPartsMultAtLeast_triangular_eq_nonnegCrank (m n : ℕ)
    (hm : 0 < m) :
    ∑ j ∈ (Finset.range (n + 1)).filter (fun j => j * (j + 1) / 2 ≤ n),
        (-1 : ℤ) ^ j * (sumDistinctPartsMultAtLeast m (n - j * (j + 1) / 2) : ℤ)
      = ∑ j ∈ (Finset.range (n + 1)).filter (fun j => m * j ≤ n),
          (j : ℤ) *
            ((Finset.univ.filter fun μ : Nat.Partition (n - m * j) =>
              0 ≤ partitionCrank μ).card : ℤ)
Sources
  • Cristina Ballantine, Combinatorial Proofs of Merca's Identities Involving the Sum of Different Parts Congruent to r Modulo m in All Partitions of n, INTEGERS 24 (2024), Article A15, Theorem 12 (i).
Uses 2
Statement usesCrank of a partition · Sum of different parts appearing at least \(m\) times
7 fully proved14 ready to prove

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