Reduced unicellular hypermonopoles

DefinitionCyclic product numbers#✓fully proved

For natural numbers \(n\) and \(k\), write \(c(m, j)\) for the unsigned Stirling numbers of the first kind. The cyclic product number \(H(n, k)\) is the quotient \(c(n + 1, k) / \binom{n + 1}{2}\), rounded down to a natural number, when \(n + k\) is even (equivalently, \(n - k\) is even), and \(0\) when \(n + k\) is odd. Division by zero gives \(0\), so \(H(0, k) = 0\) for every \(k\).

These are the numbers displayed in Theorem 5 of the source. The numbers \(H(n + 1, k)\) are the Hultman numbers (OEIS A164652).

public def cyclicProductNumber (n k : ℕ) : ℕ :=
  if Even (n + k) then Nat.stirlingFirst (n + 1) k / (n + 1).choose 2 else 0
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Theorem 5.
Used by 6
Used byCounting reduced unicellular hypermonopoles with a cyclic vertex permutation · Unicellular hypermonopoles with a cyclic vertex permutation · Vanishing of the cyclic product numbers for odd \(n - k\) · Small cyclic product numbers · Cyclic product numbers with second argument zero · Generating polynomial of the cyclic product numbers

TheoremVanishing of the cyclic product numbers for odd \(n - k\)#✓fully proved

For natural numbers \(n\) and \(k\) such that \(n + k\) is odd (equivalently, \(n - k\) is odd), the cyclic product number \(H(n, k)\) is \(0\).

public theorem cyclicProductNumber_of_odd {n k : ℕ} (h : ¬ Even (n + k)) :
    cyclicProductNumber n k = 0
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Theorem 5.
Uses 1
Statement usesCyclic product numbers

TheoremCyclic product numbers with second argument zero#✓fully proved

For every natural number \(n\), the cyclic product number \(H(n, 0)\) is \(0\), since the unsigned Stirling number of the first kind \(c(n + 1, 0)\) is \(0\).

public theorem cyclicProductNumber_zero_right (n : ℕ) : cyclicProductNumber n 0 = 0
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Theorem 5.
Uses 1
Statement usesCyclic product numbers

TheoremSmall cyclic product numbers#✓fully proved

The cyclic product numbers satisfy \(H(3, 1) = 1\), \(H(4, 2) = 5\), \(H(5, 1) = 8\) and \(H(7, 1) = 180\).

public theorem cyclicProductNumber_small_values :
    cyclicProductNumber 3 1 = 1 ∧ cyclicProductNumber 4 2 = 5 ∧
      cyclicProductNumber 5 1 = 8 ∧ cyclicProductNumber 7 1 = 180
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Table 1 (values of H(n, k)).
Uses 1
Statement usesCyclic product numbers

TheoremGenus bounds for reduced unicellular hypermonopoles#●ready to prove

Let \(\sigma\) and \(\alpha\) be permutations of \(\{0, 1, \dots, n - 1\}\), and write \(z(\pi)\) for the number of cycles of a permutation \(\pi\), fixed points included. Suppose that \(z(\sigma) = 1\), that \(z(\alpha^{-1} \circ \sigma) = 1\), that \(\alpha\) has no fixed point, and that the natural number \(g\) satisfies \(n + 2 = z(\sigma) + z(\alpha) + z(\alpha^{-1} \circ \sigma) + 2g\). Then \(2g + 1 \le n \le 4g\).

In the terms of the source, \((\sigma, \alpha)\) is a reduced unicellular hypermonopole on \(n\) points and \(g\) is its genus, given by Jacques' formula.

public theorem_wanted genus_bounds_of_reduced_unicellular_hypermonopole {n g : ℕ}
    (σ α : Equiv.Perm (Fin n)) (hσ : cycleQuotientCount σ = 1)
    (hface : cycleQuotientCount (α⁻¹ * σ) = 1) (hred : ∀ i, α i ≠ i)
    (hgenus : n + 2 = cycleQuotientCount σ + cycleQuotientCount α +
      cycleQuotientCount (α⁻¹ * σ) + 2 * g) :
    2 * g + 1 ≤ n ∧ n ≤ 4 * g
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Lemma 2 (jis-2025-25.3.3-lem-2).
Uses 1
Statement usesNumber of cycles of a permutation

TheoremGenerating polynomial of the cyclic product numbers#●ready to prove

For every natural number \(n\), let \(H(n, k)\) be the cyclic product numbers, viewed as rational numbers. Then in the polynomial ring \(\mathbb{Q}[x]\)

\[\sum_{k=0}^{n} H(n, k) x^k = c_n (x (x + 1) \cdots (x + n) - x (x - 1) \cdots (x - n)),\]

where \(c_n = \frac{1}{(n + 1) n}\) for \(n \ge 1\) and \(c_0 = 0\); for \(n = 0\) both sides are \(0\).

public theorem_wanted sum_cyclicProductNumber_mul_X_pow (n : ℕ) :
    ∑ k ∈ Finset.range (n + 1), C (cyclicProductNumber n k : ℚ) * X ^ k =
      C (((n + 1 : ℚ) * n)⁻¹) * (ascPochhammer ℚ (n + 1) - descPochhammer ℚ (n + 1))
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Lemma 6 (jis-2025-25.3.3-lem-6), quoting S. Grusea and A. Labarre.
Uses 1
Statement usesCyclic product numbers

TheoremUnicellular hypermonopoles with a cyclic vertex permutation#●ready to prove

Let \(n\) and \(k\) be natural numbers, let \(\sigma\) be the cyclic permutation \(i \mapsto i + 1\) (modulo \(n\)) of \(\{0, 1, \dots, n - 1\}\), and write \(z(\pi)\) for the number of cycles of a permutation \(\pi\), fixed points included. Then the number of permutations \(\alpha\) of \(\{0, 1, \dots, n - 1\}\) with \(z(\alpha^{-1} \circ \sigma) = 1\) and \(z(\alpha) = k\) is the cyclic product number \(H(n, k)\).

In the terms of the source, these \(\alpha\) are the unicellular hypermonopoles \((\sigma, \alpha)\) with \(\sigma = (1, \dots, n)\) and \(z(\alpha) = k\).

public theorem_wanted card_unicellular_hypermonopoles_finRotate (n k : ℕ) :
    Fintype.card {α : Equiv.Perm (Fin n) //
      cycleQuotientCount (α⁻¹ * finRotate n) = 1 ∧ cycleQuotientCount α = k} =
      cyclicProductNumber n k
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Corollary 8 (jis-2025-25.3.3-cor-8).
Uses 2
Statement usesNumber of cycles of a permutation · Cyclic product numbers

TheoremCriterion for a hypermap with a cyclic vertex permutation to have no bud#●ready to prove

Let \(\sigma\) be the cyclic permutation \(i \mapsto i + 1\) (modulo \(n\)) of \(\{0, 1, \dots, n - 1\}\). A permutation \(\alpha\) of \(\{0, 1, \dots, n - 1\}\) has no fixed point if and only if \(\pi = \alpha^{-1} \circ \sigma\) satisfies \(\pi(i) \ne i + 1\) (modulo \(n\)) for every \(i\).

In the terms of the source, a bud of the hypermap \((\sigma, \alpha)\) is a fixed point of \(\alpha\). The source assumes the hypermap is a unicellular hypermonopole; the equivalence does not use that assumption, so it is stated for every \(\alpha\).

public theorem_wanted reduced_iff_forall_face_ne_finRotate {n : ℕ}
    (α : Equiv.Perm (Fin n)) :
    (∀ i, α i ≠ i) ↔ ∀ i, (α⁻¹ * finRotate n) i ≠ finRotate n i
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Lemma 10 (jis-2025-25.3.3-lem-10).

TheoremCycle count of a fixed-point-free permutation#●ready to prove

Let \(n \ge 1\) and let \(\alpha\) be a permutation of \(\{0, 1, \dots, n - 1\}\) with no fixed point. Then the number \(z(\alpha)\) of cycles of \(\alpha\) satisfies \(1 \le z(\alpha)\) and \(2 z(\alpha) \le n\).

This is the source's bound \(1 \le z(\alpha) \le n / 2\) for a reduced unicellular hypermonopole \((\sigma, \alpha)\) on \(n\) points: only \(n \ge 1\) (forced by \(z(\sigma) = 1\)) and the absence of buds (fixed points of \(\alpha\)) are used.

public theorem_wanted cycleCount_bounds_of_reduced {n : ℕ} (hn : 1 ≤ n)
    (α : Equiv.Perm (Fin n)) (hred : ∀ i, α i ≠ i) :
    1 ≤ cycleQuotientCount α ∧ 2 * cycleQuotientCount α ≤ n
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Corollary 11 (jis-2025-25.3.3-cor-11).
Uses 1
Statement usesNumber of cycles of a permutation

TheoremCounting reduced unicellular hypermonopoles with a cyclic vertex permutation#●ready to prove

Let \(n \ge 2\) and \(k \ge 1\) be natural numbers with \(2k \le n\), let \(\sigma\) be the cyclic permutation \(i \mapsto i + 1\) (modulo \(n\)) of \(\{0, 1, \dots, n - 1\}\), and write \(z(\pi)\) for the number of cycles of a permutation \(\pi\), fixed points included. Let \(r(n, k)\) be the number of permutations \(\alpha\) of \(\{0, 1, \dots, n - 1\}\) with no fixed point such that \(z(\alpha^{-1} \circ \sigma) = 1\) and \(z(\alpha) = k\). Then, in terms of the cyclic product numbers \(H\),

\[r(n, k) = \sum_{i=0}^{k-1} (-1)^i \binom{n}{i} H(n - i, k - i).\]

In the terms of the source, \(r(n, k)\) counts the reduced unicellular hypermonopoles \((\sigma, \alpha)\) with \(\sigma = (1, \dots, n)\) and \(z(\alpha) = k\).

public theorem_wanted card_reduced_unicellular_hypermonopoles_finRotate {n k : ℕ}
    (hn : 2 ≤ n) (hk : 1 ≤ k) (hkn : 2 * k ≤ n) :
    (Fintype.card {α : Equiv.Perm (Fin n) //
      cycleQuotientCount (α⁻¹ * finRotate n) = 1 ∧ (∀ i, α i ≠ i) ∧
        cycleQuotientCount α = k} : ℤ) =
      ∑ i ∈ Finset.range k,
        (-1) ^ i * (n.choose i : ℤ) * cyclicProductNumber (n - i) (k - i)
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Proposition 12 (jis-2025-25.3.3-prop-12).
Uses 2
Statement usesNumber of cycles of a permutation · Cyclic product numbers

TheoremVanishing of the reduced unicellular hypermonopole counts#●ready to prove

Let \(\sigma_n\) be the cyclic permutation \(i \mapsto i + 1\) (modulo \(n\)) of \(\{0, 1, \dots, n - 1\}\), write \(z(\pi)\) for the number of cycles of a permutation \(\pi\), fixed points included, and let \(r(n, k)\) be the number of permutations \(\alpha\) of \(\{0, 1, \dots, n - 1\}\) with no fixed point such that \(z(\alpha^{-1} \circ \sigma_n) = 1\) and \(z(\alpha) = k\). Then \(r(n, k) = 0\) for all natural numbers \(n\) and \(k\) with \(n - k\) odd; \(r(3, k) > 0\) for some natural number \(k\) with \(2k \le 3\); and \(r(n, k) = 0\) whenever \(n < 3\) and \(2k \le n\).

So the least \(n\) with \(r(n, k) > 0\) for some \(k \le n / 2\) is \(n = 3\). In the terms of the source, \(r(n, k)\) counts the reduced unicellular hypermonopoles \((\sigma_n, \alpha)\) with \(\sigma_n = (1, \dots, n)\) and \(z(\alpha) = k\).

public theorem_wanted reduced_unicellular_hypermonopoles_finRotate_vanishing :
    (∀ n k : ℕ, Odd ((n : ℤ) - k) →
      Fintype.card {α : Equiv.Perm (Fin n) //
        cycleQuotientCount (α⁻¹ * finRotate n) = 1 ∧ (∀ i, α i ≠ i) ∧
          cycleQuotientCount α = k} = 0) ∧
    (∃ k : ℕ, 2 * k ≤ 3 ∧
      0 < Fintype.card {α : Equiv.Perm (Fin 3) //
        cycleQuotientCount (α⁻¹ * finRotate 3) = 1 ∧ (∀ i, α i ≠ i) ∧
          cycleQuotientCount α = k}) ∧
    ∀ n < 3, ∀ k : ℕ, 2 * k ≤ n →
      Fintype.card {α : Equiv.Perm (Fin n) //
        cycleQuotientCount (α⁻¹ * finRotate n) = 1 ∧ (∀ i, α i ≠ i) ∧
          cycleQuotientCount α = k} = 0
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Corollary 13 (jis-2025-25.3.3-cor-13).
Uses 1
Statement usesNumber of cycles of a permutation

TheoremCounting reduced unicellular hypermonopoles by genus#●ready to prove

Fix \(g \in \mathbb{N}\). For each \(n \in \mathbb{N}\) let \(\zeta_n\) be the cyclic permutation of \(\{0, \ldots, n - 1\}\) sending \(i\) to \(i + 1\) modulo \(n\), and let \(z(\pi)\) be the number of cycles of a permutation \(\pi\), fixed points included. Let \(U_g\) be the set of pairs \((n, \alpha)\), with \(n \in \mathbb{N}\) and \(\alpha\) a permutation of \(\{0, \ldots, n - 1\}\), such that \(z(\zeta_n) = 1\), \(z(\alpha^{-1} \zeta_n) = 1\) (the product applies \(\zeta_n\) first), \(\alpha\) has no fixed point, and \(n + 2 = z(\zeta_n) + z(\alpha) + z(\alpha^{-1} \zeta_n) + 2g\). Then \(U_g\) is finite and its number of elements is \(\sum_{n = 2g + 1}^{4g} r(n, n - 2g)\), where \(r(n, k)\) is the number of permutations \(\alpha\) of \(\{0, \ldots, n - 1\}\) without fixed points such that \(z(\alpha^{-1} \zeta_n) = 1\) and \(z(\alpha) = k\).

The pairs in \(U_g\) are the reduced unicellular hypermonopoles \((\zeta_n, \alpha)\) of genus \(g\), with the genus given by Jacques' formula and the vertex permutation normalized to \(\zeta_n\). The paper writes \(u(g)\) for the number of elements of \(U_g\); its first value is \(u(1) = 2\).

public theorem_wanted reduced_unicellular_hypermonopoles_of_genus_count (g : ℕ) :
    let U : Set (Σ n : ℕ, Equiv.Perm (Fin n)) :=
      {p | cycleQuotientCount (finRotate p.1) = 1 ∧
        cycleQuotientCount (p.2⁻¹ * finRotate p.1) = 1 ∧ (∀ i, p.2 i ≠ i) ∧
        p.1 + 2 = cycleQuotientCount (finRotate p.1) + cycleQuotientCount p.2 +
          cycleQuotientCount (p.2⁻¹ * finRotate p.1) + 2 * g}
    U.Finite ∧
      U.ncard = ∑ n ∈ Finset.Icc (2 * g + 1) (4 * g),
        Fintype.card {α : Equiv.Perm (Fin n) //
          cycleQuotientCount (α⁻¹ * finRotate n) = 1 ∧ (∀ i, α i ≠ i) ∧
            cycleQuotientCount α = n - 2 * g}
Sources
  • Robert Cori and Gábor Hetyei, On Reduced Unicellular Hypermonopoles, Journal of Integer Sequences 28 (2025), Article 25.3.3, Proposition 15 (jis-2025-25.3.3-prop-15).
Uses 1
Statement usesNumber of cycles of a permutation
4 fully proved8 ready to prove

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