bibkey: ishikawa2012holonomic authors: Masao Ishikawa, Christoph Koutschan year: 2012 title: “Zeilberger’s Holonomic Ansatz for Pfaffians” doi: 10.48550/arXiv.1201.5253 url: https://arxiv.org/abs/1201.5253v2 claim: “The paper proves by the holonomic ansatz that Pf((j-i) M_{i+j-3}){1<=i,j<=2n} equals the product of 4k+1 over k < n for the Motzkin numbers M_n, and conjectures product formulas for the Pfaffians Pf((j-i) M^(k){i+j-2}) of the columns of the Motzkin triangle and of sums of two consecutive entries.” strata_touched:

• D5/S0/Certificates/IshikawaKoutschanMotzkinPfaffianRefutation license: citation-only triage: anchor

Zeilberger’s holonomic ansatz for Pfaffians

Ishikawa and Koutschan adapt Zeilberger’s holonomic ansatz from determinants to Pfaffians. For a 2n × 2n skew-symmetric matrix A = (a_{i,j}) the section on Pfaffians defines

Pf(A) = Σ ε(σ₁, σ₂, …, σ_{2n−1}, σ_{2n}) a_{σ₁σ₂} ⋯ a_{σ_{2n−1}σ_{2n}},

where the summation is over all partitions {{σ₁,σ₂},…,{σ_{2n−1},σ_{2n}}} of [2n] into two-element subsets and ε is the sign of the permutation (1 2 ⋯ 2n ↦ σ₁ σ₂ ⋯ σ_{2n}). With the Motzkin numbers M_n, which count the paths from (0,0) to (n,0) with steps U = (1,1), H = (1,0), D = (1,−1) that never run below the horizontal axis, Theorem thm.pfMotz states

Pf((j−i) M_{i+j−3}){1≤i,j≤2n} = ∏{k=0}^{n−1}(4k+1) for all n ≥ 1.

The closing section writes 𝓜^{(k)}_i = h(i,2k−1) for the number of Motzkin paths from (0,0) to (i−1,k−1), so that M_n = 𝓜^{(1)}_{n+1}, and states Conjecture conj.gen: for positive integers n and k, part (i),

Pf((j−i)𝓜^{(k)}{i+j−2}){1≤i,j≤2n} equals ∏{i=0}^{m−1}∏{j=0}^{k−1}(4ki+2j+k) if m = n/k is an integer, and it equals (∏{j=1}^{⌊k/2⌋} 1/(2j−k))(∏{i=0}^{m−1}∏_{j=1}^{k}(4ki+2j−k)) if k is odd and m = (n+⌊k/2⌋)/k is an integer. The Pfaffian is zero in all other cases.

and a part (ii) for the entries 𝓜^{(k)}_{i+j−2} + 𝓜^{(k)}_{i+j−1}. The authors remark that part (i) at k = 1 is Theorem thm.pfMotz and that their method does not apply for k ≥ 2, because the Pfaffians vanish periodically.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1201.5253
• URL: https://arxiv.org/abs/1201.5253v2
• Version and location: arXiv:1201.5253v2 (2012-05-16), source file pfaffians.tex: the Pfaffian definition in the section “Pfaffians”, Theorem thm.pfMotz in the section “A Motzkin Number Pfaffian”, and Conjecture conj.gen at the end of the section “Application of Theorem 2”.