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Subassociative groupoids

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subassociative-groupoidsCombinatoricsmath.COposed by Milton S. Braitt, Donald Silbergerrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

Specify the size sequences of ⟨ℤ; -⟩ for each k ≥ 2.

Context

Candidate 1 of the open problems stated in "Subassociative groupoids", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
    AI involvement
    ai discovered
    — the result was found by a model.
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    GPT-5.5 xhigh

    NEW

    Problem: For k≥2k\ge2, let Tk\mathcal T_k be the Catalan set of all parenthesizations of

    x1−x2−⋯−xkx_1-x_2-\cdots-x_k

    with the variables kept in this order. Define P∼QP\sim Q if the two bracketed subtraction terms induce the same kk-ary term operation on Z\mathbb Z. The requested size sequence is the multiset, usually sorted increasingly, of the ∼\sim-class sizes.

    Result: For a binary word w=w1⋯wmw=w_1\cdots w_m, write wˉ\bar w for bitwise complement and define N(∅)=1N(\varnothing)=1. For m≥1m\ge1, put

    a0=1,ai=wi (1≤i≤m),am+1=0,a_0=1,\qquad a_i=w_i\ (1\le i\le m),\qquad a_{m+1}=0,

    and define

    N(w)=∑0≤i≤mai=1, ai+1=0N(w1⋯wi−1) N ⁣(wi+2⋯wm‾),N(w)= \sum_{\substack{0\le i\le m\\ a_i=1,\ a_{i+1}=0}} N(w_1\cdots w_{i-1})\, N\!\left(\overline{w_{i+2}\cdots w_m}\right),

    with empty subwords allowed.

    Then for (Z,−)(\mathbb Z,-), the equivalence classes in arity kk are indexed by the 2k−22^{k-2} words w∈{0,1}k−2w\in\{0,1\}^{k-2}. The class indexed by ww consists exactly of the parenthesizations evaluating to

    x1−x2+∑j=3k(−1)wj−2xj,x_1-x_2+\sum_{j=3}^k (-1)^{w_{j-2}}x_j,

    and its size is N(w)N(w). Hence the required size sequence is

    sort⁡{ N(w):w∈{0,1}k−2 }.\operatorname{sort}\{\,N(w):w\in\{0,1\}^{k-2}\,\}.

    For example:

    k=2: (1),k=2:\ (1), k=3: (1,1),k=3:\ (1,1), k=4: (1,1,1,2),k=4:\ (1,1,1,2), k=5: (1,1,1,1,2,2,3,3),k=5:\ (1,1,1,1,2,2,3,3), k=6: (1,1,1,1,1,2,2,2,3,3,3,3,4,4,4,7).k=6:\ (1,1,1,1,1,2,2,2,3,3,3,3,4,4,4,7).

    Proof sketch: a parenthesized subtraction term is a linear form

    ∑j=1kϵjxj,ϵj∈{±1},\sum_{j=1}^k \epsilon_jx_j,\qquad \epsilon_j\in\{\pm1\},

    where ϵj=(−1)rj\epsilon_j=(-1)^{r_j} and rjr_j is the number of right edges on the path to leaf jj in the binary syntax tree. Always ϵ1=+1\epsilon_1=+1 and ϵ2=−1\epsilon_2=-1. Distinct sign vectors give distinct term functions on Z\mathbb Z, by testing a single variable.

    At the root, if the left subtree has i+1i+1 leaves, the first leaf of the right subtree has odd right-depth parity and the next has even parity; equivalently, the augmented word 1w01w0 has a descent 1010 at position ii. The left subtree contributes N(w1⋯wi−1)N(w_1\cdots w_{i-1}), while the right subtree contributes N(wi+2⋯wm‾)N(\overline{w_{i+2}\cdots w_m}), because entering the right subtree flips all parities. Summing over all descents gives exactly the recurrence above.

    Citation: No external result is needed; the formula and proof above are self-contained.

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The claimed solution attacks the correct statement and gives a rigorous recursive specification of all class sizes. The sign-vector characterization is correct for subtraction, distinct sign vectors give distinct integer term functions, and the root-split argument gives a bijective recurrence: descents of 1w01w0 are exactly possible split positions, with the right suffix complemented because entering the right subtree flips signs. This fully specifies the sorted size sequence for every kk. I found no existing stronger exact size-sequence formula in the cited/recent literature.

      Novelty assessment

      TYPE1

      Classification rationale: The result appears genuinely new as an exact recursive specification of all fiber sizes for integer subtraction, but it is a very small refinement of the already-known right-depth/parity characterization of subtraction bracketings. The proof is a standard root-splitting recurrence for binary trees and is likely too short/routine to support a standalone paper in a standard combinatorics journal, though it could be a useful note or a small section in a broader paper.

      Literature check: Closest known results: Csákány–Waldhauser and later Hein–Huang/Huang–Lehtonen establish the associative spectrum of subtraction as 2n−22^{n-2}, equivalently classify bracketings by right-depth parity/sign vector. Hein–Huang’s “Modular Catalan Numbers” also studies equivalence-class sizes in the broader kk-associative setting, including largest class sizes, but I found no full size-sequence formula or recurrence matching this result. OEIS searches for the displayed size rows and relevant phrases (“subassociative groupoids”, “right depths modulo”, “k-associative equivalence size”) did not reveal the exact sequence.

      Citation: Closest references: M. S. Braitt and D. Silberger, “Subassociative groupoids,” Quasigroups Related Systems 14 (2006), 11–26, Problem 3.8; N. Hein and J. Huang, “Modular Catalan Numbers,” European J. Combin. 61 (2017), 197–218; J. Huang and E. Lehtonen, “Associative-commutative spectra for some varieties of groupoids,” Enumer. Combin. Appl. 4(4) (2024), Art. S2R29.

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