slug: oeis-a398542-fixed-bottom-polynomial bibkey: norton2026a398542 doi: null url: https://oeis.org/A398542 triage: theorem motivation_gids:

• D5/S1/Words/Patterns/DerangementRatioNonconvergence.Contains

A398542 fixed-bottom polynomial conjecture

Problem

Charles Cornell Norton, OEIS A398542, August 1, 2026; revision 18, August 30, 2026, 18:45:04, exact selected %F:

Conjecture: for every m >= 1 and every b in S(m), d(b,k) = p(k)binomial(2k,k) + q(k)*4^k for polynomials p and q depending on b, with deg p <= m-1 and deg q <= m-2. Verified for m <= 6 and k <= 12.

S(m)=Av_m(132). For every m>=1 and b in S(m), choose p_b,q_b in Q[X] independently of k such that the displayed identity holds for every k>=0. The degree bounds are m-1 and m-2 for m>=2, with q_b=0 when m=1. Here d(b,k) counts actual permutations of 1,…,m+k avoiding 1324 whose lower-value subsequence is exactly b and whose standardized upper-value subsequence avoids 213. The value cut is fixed, the empty upper cell is allowed, and d(b,0)=1. Rational coefficients are an explicit sufficient choice beyond the source’s unspecified coefficient field.

Motivation

The preregistration https://github.com/the-omega-institute/trureturing/issues/9332 contains the exact source and full ordinary proof. The frozen generic Contains definition supplies classical pattern containment by increasing position embeddings and exact relative value order. The target is the whole fixed-bottom assertion, not a bounded fit or a recursively defined count.

Gap

The source’s checked range m<=6, k<=12 does not imply its unbounded conjecture. Aggregate domino enumeration does not determine every fixed bottom. The missing uniform connection is an actual minimum-split equivalence followed by a guarded inverse-power bound for all intervals. The new Library note records the bounded prior-source comparison. No exact settlement was found in those inspected bodies; worldwide priority is not claimed.

Route

Insert upper permutations into weakly increasing gaps of the same complete bottom b. Mixed 1324 occurrences are exactly interleaved lower and upper ascents, giving the strict deadline cutoff. Split at the upper minimum, with the forced value shifts and either child allowed to be empty. Take cardinalities of the actual equivalence to obtain the interval convolution. In Q[[X]], separate singleton Catalan states and prove F(l,h)=T A(T), deg A<=2(h-l)-2, only for l<h<=m and h<dead(l), where T=(1-2XC)^(-1). Extract odd/even powers through the pinned binomial and Pochhammer identities to obtain the rational polynomials and degree bounds.

Falsifier

Removing the guard is invalid: for b=(1,2), dead(1)=2 and F(1,2)=C^2=4/(1+s)^2, where s=1-2XC. This interval fails h<dead(l); the full state F(0,2)=(1+s)/(2s^2) is guarded. Removing the strict cutoff counts six rather than five objects for this bottom, interval [1,2], and upper size two. These are method falsifiers, not counterexamples to the source conjecture. Replacing actual cardinality or allowing p,q to depend on k would fail the stated target.

Evidence

A398542FixedBottom.Actual defines the literal permutation subtype and actualEquiv proves its configuration equivalence. A398542MinimumRecurrence.actual_cardinal_recurrence supplies the actual minimum split, zero coefficient and full-interval equality. A398542Polynomial.actual_series and guarded_polynomial connect that cardinality to the formal series argument. A398542Polynomial.result states the complete all-m, all-b, all-k conclusion with rational polynomials, both bounds and explicit m=1 zero condition. Its Nat.centralBinom is Nat.choose (2*k) k. Admission and resolution state are supplied by the canonical report, Freeze and typed Scribe claim, not by a handwritten status in this dossier.

Triage

theorem: first-tier recent fixed-bottom conjecture, allocated on the strength of the complete ordinary proof in #9332. The historical tier-3/note-only record in Library/Words/oeis2026triage0911.md remains unchanged. This allocation does not reclassify unrestricted 1324 enumeration. Neither A398446 nor the full L-gridding generating function is settled by this statement; supporting formalizations are not additional resolved problems.

ASSUMED-UNVERIFIED

Worldwide priority and the absence of a proof outside the inspected literature are unverified. The bounded source audit is attributed in norton2026a398542; this delivery directly rechecked OEIS revision 18. Source fidelity is a semantic review obligation beyond kernel type checking. Correctness of the separate official program, the optional tree representation, unrestricted enumeration, publication, required CI, merge and the caller’s completion audit are separate from this theorem.