Toeplitz–Hessenberg determinants of Fuss–Catalan numbers

TheoremSmall generalized Fuss–Catalan numbers#✓fully proved

For the generalized Fuss–Catalan numbers \(A_n(p, r) = \frac{r}{pn + r} \binom{pn + r}{n}\), one has \(A_3(2, 1) = 5\), \(A_2(2, 3) = 9\), \(A_3(3, 1) = 12\) and \(A_2(3, 2) = 7\).

With \(C_n\) the \(n\)-th Catalan number, \(A_3(2, 1) = C_3\) and \(A_2(2, 3) = C_4 - C_3\).

theorem toeplitzHessenberg_fussCatalan_examples :
    generalizedFussCatalanNumber 2 3 1 = 5 ∧ generalizedFussCatalanNumber 2 2 3 = 9 ∧
      generalizedFussCatalanNumber 3 3 1 = 12 ∧ generalizedFussCatalanNumber 3 2 2 = 7

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Sections 1-2 (C_n = A_n(2, 1), K_n = A_n(2, 3), H_n = A_n(3, 1), J_n = A_n(3, 2)).

Uses 1

Statement usesGeneralized Fuss–Catalan number

TheoremGenerating function of Toeplitz–Hessenberg determinants#●ready to prove

Let \(R\) be a commutative ring and \(a_0, a_1, a_2, \dots\) a sequence in \(R\). Write \(\det(a_0; a_1, \dots, a_n)\) for the determinant of the \(n \times n\) Toeplitz–Hessenberg matrix whose entry in row \(i\) and column \(j\) is \(a_{i+1-j}\) for \(j \le i + 1\) and \(0\) above the superdiagonal. In the ring of formal power series over \(R\) let \(f(x) = \sum_{n \ge 1} \det(a_0; a_1, \dots, a_n)\, x^n\) and \(H(x) = \sum_{i \ge 1} a_i (-a_0)^{i-1} x^i\). Then \(f = H + H f\).

When \(a_0\) is invertible, \(H(x) = -g(-a_0 x)/a_0\) with \(g(x) = \sum_{i \ge 1} a_i x^i\), and the identity is the paper's \(f(x) = (-g(-a_0 x)/a_0)/(1 + g(-a_0 x)/a_0)\). The form \(f = H + H f\) needs no hypothesis on \(a_0\); for \(a_0 = 0\) it gives \(\det(0; a_1, \dots, a_n) = a_1^n\).

public theorem_wanted toeplitzHessenbergDet_genFun {R : Type*} [CommRing R] (a : ℕ → R) :
    PowerSeries.mk (fun n => if n = 0 then 0 else toeplitzHessenbergDet a n) =
      PowerSeries.mk (fun i => if i = 0 then 0 else a i * (-a 0) ^ (i - 1)) +
        PowerSeries.mk (fun i => if i = 0 then 0 else a i * (-a 0) ^ (i - 1)) *
          PowerSeries.mk (fun n => if n = 0 then 0 else toeplitzHessenbergDet a n)

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Lemma 2 (jis-2026-26.1.2-lem-2).

Uses 1

Statement usesToeplitz–Hessenberg determinant

TheoremToeplitz–Hessenberg determinants of Fuss–Catalan numbers#●ready to prove

Write \(\det(a_0; a_1, \dots, a_n)\) for the determinant of the \(n \times n\) Toeplitz–Hessenberg matrix whose entry in row \(i\) and column \(j\) is \(a_{i+1-j}\) for \(j \le i + 1\) and \(0\) above the superdiagonal. For natural numbers \(p, r, n\) let \(A_n(p, r) = \frac{r}{pn + r} \binom{pn + r}{n}\) be the generalized Fuss–Catalan number, taken to be \(0\) when \(pn + r = 0\). Then for all natural numbers \(p, r\) and every \(n \ge 1\),

\[\det(1; A_1(p, r), \dots, A_n(p, r)) = (-1)^{n-1} \frac{r}{n} \binom{pn - r - 1}{n - 1}\]

in \(\mathbb{Q}\), where the binomial coefficient has an integer, possibly negative, upper index: \(\binom{x}{k} = x (x - 1) \cdots (x - k + 1)/k!\).

This includes \(p = r = 0\), which the paper's standing assumption excludes; there every entry \(A_i(0, 0)\) is \(0\) and both sides vanish.

public theorem_wanted toeplitzHessenbergDet_fussCatalan (p r n : ℕ) (hn : 1 ≤ n) :
    toeplitzHessenbergDet
        (fun i => if i = 0 then 1 else (generalizedFussCatalanNumber p i r : ℚ)) n =
      (-1) ^ (n - 1) * (r : ℚ) / n * (Ring.choose ((p * n : ℤ) - r - 1) (n - 1) : ℚ)

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Theorem 3 (jis-2026-26.1.2-thm-3).

Uses 2

Statement usesGeneralized Fuss–Catalan number · Toeplitz–Hessenberg determinant

TheoremSpecial Toeplitz–Hessenberg determinants of Fuss–Catalan numbers#●ready to prove

Write \(\det(a_0; a_1, \dots, a_n)\) for the determinant of the \(n \times n\) Toeplitz–Hessenberg matrix whose entry in row \(i\) and column \(j\) is \(a_{i+1-j}\) for \(j \le i + 1\) and \(0\) above the superdiagonal. For natural numbers \(p, r \ge 1\) let \(A_n(p, r) = \frac{r}{pn + r} \binom{pn + r}{n}\) be the generalized Fuss–Catalan number, put \(D_n(p, r) = (-1)^{n-1} \det(1; A_1(p, r), \dots, A_n(p, r))\), and let \(C_n\) be the \(n\)-th Catalan number. Then, in \(\mathbb{Q}\): \(D_n(2, 3) = C_{n-1} - C_{n-2}\) for every \(n \ge 2\); \(D_n(2, 4) = C_{n-1} - 2 C_{n-2}\) for every \(n \ge 3\); \(D_n(3, 1) = \frac{1}{n} \binom{3n - 2}{n - 1}\) for every \(n \ge 1\); and \(D_n(3, 2) = \frac{2}{n} \binom{3n - 3}{n - 1}\) for every \(n \ge 1\).

MetaMathlibExt.toeplitzHessenbergDet_fussCatalan_special · 17 lines

public theorem_wanted toeplitzHessenbergDet_fussCatalan_special :
    (∀ n : ℕ, 2 ≤ n →
      (-1) ^ (n - 1) * toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (generalizedFussCatalanNumber 2 i 3 : ℚ)) n =
        (catalan (n - 1) : ℚ) - catalan (n - 2)) ∧
    (∀ n : ℕ, 3 ≤ n →
      (-1) ^ (n - 1) * toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (generalizedFussCatalanNumber 2 i 4 : ℚ)) n =
        (catalan (n - 1) : ℚ) - 2 * catalan (n - 2)) ∧
    (∀ n : ℕ, 1 ≤ n →
      (-1) ^ (n - 1) * toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (generalizedFussCatalanNumber 3 i 1 : ℚ)) n =
        1 / (n : ℚ) * ((3 * n - 2).choose (n - 1) : ℚ)) ∧
    ∀ n : ℕ, 1 ≤ n →
      (-1) ^ (n - 1) * toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (generalizedFussCatalanNumber 3 i 2 : ℚ)) n =
        2 / (n : ℚ) * ((3 * n - 3).choose (n - 1) : ℚ)

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Corollary 4 (jis-2026-26.1.2-cor-4).

Uses 2

Statement usesGeneralized Fuss–Catalan number · Toeplitz–Hessenberg determinant

TheoremToeplitz–Hessenberg determinants with Catalan entries#●ready to prove

Write \(\det(a_0; a_1, \dots, a_n)\) for the determinant of the \(n \times n\) Toeplitz–Hessenberg matrix whose entry in row \(i\) and column \(j\) is \(a_{i+1-j}\) for \(j \le i + 1\) and \(0\) above the superdiagonal (so \(a_0\) fills the superdiagonal and \(a_1, \dots, a_n\) the first column); for \(n = 0\) it is \(1\). Let \(C_n\) be the \(n\)-th Catalan number, \(C(x) = \sum_{n \ge 0} C_n x^n \in \mathbb{Q}[[x]]\), and \(K_m = C_{m+2} - C_{m+1}\) for \(m \ge -1\), so that \(K_{-1} = 0\). Then the following hold. (1) \(\sum_{n \ge 1} \det(-1; C_1, C_3, \dots, C_{2n-1})\, x^{2n} = x^2 C(x)^2 C(-x)^2\). (2) \(\bigl(\sum_{n \ge 0} \det(-1; C_2, \dots, C_{n+1})\, x^n\bigr) \bigl(2 - C(x)^2\bigr) = 1\). (3) For every \(n \ge 1\), \(\det(1; K_{-1}, K_0, \dots, K_{n-2}) = (-1)^{n-1} \sum_{i=1}^{n-1} C_i\). (4) For every \(n \ge 1\), \(\det(1; K_0, K_1, \dots, K_{n-1}) = (-1)^{n-1} \sum_{i=0}^{n-1} C_i\). (5) \(\bigl(\sum_{n \ge 1} \det(-1; K_0, K_1, \dots, K_{n-1})\, x^n\bigr) \bigl(1 - x C(x)^3\bigr) = x C(x)^3\).

Thus the determinants in (2) have generating function \(1/(2 - C(x)^2)\) and those in (5) have generating function \(x C(x)^3/(1 - x C(x)^3)\), the generating function of OEIS A026674; the determinants in (1) form OEIS A066357.

MetaMathlibExt.toeplitzHessenbergDet_catalan · 21 lines

public theorem_wanted toeplitzHessenbergDet_catalan :
    let C : PowerSeries ℚ := PowerSeries.mk fun n => (catalan n : ℚ)
    (PowerSeries.mk (fun m => if m % 2 = 0 ∧ 2 ≤ m then
        toeplitzHessenbergDet (fun i => if i = 0 then -1 else (catalan (2 * i - 1) : ℚ)) (m / 2)
      else 0) =
      PowerSeries.X ^ 2 * C ^ 2 * (PowerSeries.rescale (-1 : ℚ) C) ^ 2) ∧
    (PowerSeries.mk (fun n =>
        toeplitzHessenbergDet (fun i => if i = 0 then -1 else (catalan (i + 1) : ℚ)) n) *
      (2 - C ^ 2) = 1) ∧
    (∀ n : ℕ, 1 ≤ n →
      toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (catalan i : ℚ) - catalan (i - 1)) n =
        (-1) ^ (n - 1) * ∑ i ∈ Finset.Icc 1 (n - 1), (catalan i : ℚ)) ∧
    (∀ n : ℕ, 1 ≤ n →
      toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (catalan (i + 1) : ℚ) - catalan i) n =
        (-1) ^ (n - 1) * ∑ i ∈ Finset.range n, (catalan i : ℚ)) ∧
    PowerSeries.mk (fun n => if n = 0 then 0 else
        toeplitzHessenbergDet
          (fun i => if i = 0 then -1 else (catalan (i + 1) : ℚ) - catalan i) n) *
      (1 - PowerSeries.X * C ^ 3) = PowerSeries.X * C ^ 3

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Theorem 6 (jis-2026-26.1.2-thm-6).

Uses 1

Statement usesToeplitz–Hessenberg determinant

TheoremToeplitz–Hessenberg determinants of ternary Fuss–Catalan numbers#●ready to prove

Write \(\det(a_0; a_1, \dots, a_n)\) for the determinant of the \(n \times n\) Toeplitz–Hessenberg matrix whose entry in row \(i\) and column \(j\) is \(a_{i+1-j}\) for \(j \le i + 1\) and \(0\) above the superdiagonal, and let \(H_n = \frac{1}{3n + 1} \binom{3n + 1}{n}\) and \(J_n = \frac{2}{3n + 2} \binom{3n + 2}{n}\). Then, in \(\mathbb{Q}\), \(\det(1; H_2, H_3, \dots, H_{n+1}) = \frac{2 (-1)^{n-1}}{n} \binom{3n - 3}{n - 1}\) for every \(n \ge 2\), and \(\det(-1; J_0, J_1, \dots, J_{n-1}) = H_n\) for every \(n \ge 1\).

public theorem_wanted toeplitzHessenbergDet_ternary :
    (∀ n : ℕ, 2 ≤ n →
      toeplitzHessenbergDet
          (fun i => if i = 0 then 1 else (generalizedFussCatalanNumber 3 (i + 1) 1 : ℚ)) n =
        2 * (-1) ^ (n - 1) / (n : ℚ) * ((3 * n - 3).choose (n - 1) : ℚ)) ∧
    ∀ n : ℕ, 1 ≤ n →
      toeplitzHessenbergDet
          (fun i => if i = 0 then -1 else (generalizedFussCatalanNumber 3 (i - 1) 2 : ℚ)) n =
        (generalizedFussCatalanNumber 3 n 1 : ℚ)

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Theorem 8, formulas (4) and (6) (jis-2026-26.1.2-thm-8).

Uses 2

Statement usesGeneralized Fuss–Catalan number · Toeplitz–Hessenberg determinant

TheoremTrudi multi-sums of Fuss–Catalan numbers#●ready to prove

For \(s \in \mathbb{Q}\) and rational numbers \(a_1, \dots, a_n\) let \(T_s(a_1, \dots, a_n)\) be the sum of \(s^{|\alpha|} \binom{|\alpha|}{\alpha_1, \dots, \alpha_n} a_1^{\alpha_1} \cdots a_n^{\alpha_n}\) over all \((\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n\) with \(\alpha_1 + 2 \alpha_2 + \dots + n \alpha_n = n\), where \(|\alpha| = \alpha_1 + \dots + \alpha_n\). Let \(A_n(p, r) = \frac{r}{pn + r} \binom{pn + r}{n}\) be the generalized Fuss–Catalan number, taken to be \(0\) when \(pn + r = 0\); let \(C_n\) be the \(n\)-th Catalan number, \(K_m = C_{m+2} - C_{m+1}\) for \(m \ge -1\) (so \(K_{-1} = 0\)), \(H_n = A_n(3, 1)\) and \(J_n = A_n(3, 2)\). Then the following hold. (1) For all natural numbers \(p, r\) and every \(n \ge 1\), \(-T_{-1}(A_1(p, r), \dots, A_n(p, r)) = \frac{r}{n} \binom{pn - r - 1}{n - 1}\), where the binomial coefficient has an integer, possibly negative, upper index: \(\binom{x}{k} = x (x - 1) \cdots (x - k + 1)/k!\). (2) For every \(n \ge 1\), \(-T_{-1}(K_{-1}, K_0, \dots, K_{n-2}) = \sum_{i=1}^{n-1} C_i\). (3) For every \(n \ge 1\), \(-T_{-1}(K_0, K_1, \dots, K_{n-1}) = \sum_{i=0}^{n-1} C_i\). (4) For every \(n \ge 2\), \(-T_{-1}(H_2, H_3, \dots, H_{n+1}) = \frac{2}{n} \binom{3n - 3}{n - 1}\). (5) For every \(n \ge 1\), \(T_1(J_0, J_1, \dots, J_{n-1}) = H_n\).

Identity (1) includes \(p = r = 0\), which the paper's standing assumption excludes; there both sides are \(0\).

MetaMathlibExt.trudiPartitionSum_fussCatalan · 16 lines

public theorem_wanted trudiPartitionSum_fussCatalan :
    (∀ p r n : ℕ, 1 ≤ n →
      -trudiPartitionSum (-1) (fun i => (generalizedFussCatalanNumber p i r : ℚ)) n =
        (r : ℚ) / n * (Ring.choose ((p * n : ℤ) - r - 1) (n - 1) : ℚ)) ∧
    (∀ n : ℕ, 1 ≤ n →
      -trudiPartitionSum (-1) (fun i => (catalan i : ℚ) - catalan (i - 1)) n =
        ∑ i ∈ Finset.Icc 1 (n - 1), (catalan i : ℚ)) ∧
    (∀ n : ℕ, 1 ≤ n →
      -trudiPartitionSum (-1) (fun i => (catalan (i + 1) : ℚ) - catalan i) n =
        ∑ i ∈ Finset.range n, (catalan i : ℚ)) ∧
    (∀ n : ℕ, 2 ≤ n →
      -trudiPartitionSum (-1) (fun i => (generalizedFussCatalanNumber 3 (i + 1) 1 : ℚ)) n =
        2 / (n : ℚ) * ((3 * n - 3).choose (n - 1) : ℚ)) ∧
    ∀ n : ℕ, 1 ≤ n →
      trudiPartitionSum 1 (fun i => (generalizedFussCatalanNumber 3 (i - 1) 2 : ℚ)) n =
        (generalizedFussCatalanNumber 3 n 1 : ℚ)

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Corollary 9 (jis-2026-26.1.2-cor-9).

Uses 2

Statement usesGeneralized Fuss–Catalan number · Trudi's partition sum

TheoremEven noncrossing partitions and Dyck paths with even descents#●ready to prove

For every natural number \(n\), there is a bijection between the noncrossing partitions of \(\{0, \dots, 2n - 1\}\) all of whose blocks have even size and the Dyck paths of semilength \(2n\) all of whose descents have even length. Here the descents of a Dyck path are the lengths of its maximal runs of down steps, one for each peak \(UD\): the run of down steps starting at the \(D\) of that peak.

public theorem_wanted nonempty_equiv_evenNoncrossing_evenDescentDyck (n : ℕ) :
    Nonempty
      ({P : _root_.NoncrossingPartition (2 * n) // ∀ B ∈ P.val.parts, Even B.card} ≃
        {p : DyckWord // p.semilength = 2 * n ∧ ∀ j ∈ dyckPeakDescents p.toList, Even j})

Sources

• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Fuss-Catalan Entries, Journal of Integer Sequences 29 (2026), Article 26.1.2, Lemma 10 (jis-2026-26.1.2-lem-10).

Uses 4

Statement usesPeak descents · Noncrossing partition · Noncrossing set partition · Set partition

1 fully proved7 ready to prove

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