slug: oeis-a193020-minus-four-eventual-identity
bibkey: layman2011a193020
doi: null
url: https://oeis.org/A193020
triage: theorem
motivation_gids:
  - D5/S1/Words/DoubleTailShuffleSupport.result

The A193020 minus-four eventual identity

Problem

OEIS A193020, revision 13, attributed to John W. Layman and the OEIS contributors, states the separate assertion a(2^k-4)=k*(k-1)/2 with its sufficiently-large lower-bound caveat. For positive n, a(n) counts distinct complete order-preserving self-shuffle output words of the ordinary binary expansion of n. Different schedules producing the same word count once.

The exact selected assertion is

$$\exists K\in \mathbb{N},3\le K\wedge \forall k\in \mathbb{N},K\le k⟹a(2^k-4)=\frac{k(k-1)}{2}.$$

The OEIS entry includes a(0). The selected event uses positive inputs throughout its tail; it does not impose a source convention at zero. This dossier anchors only the minus-four question.

Motivation

The frozen theorem D5/S1/Words/DoubleTailShuffleSupport.result proves the whole eventual assertion with K=3. An exhaustive description of the unbounded family’s distinct support separates word cardinality from the number of labelled interleaving schedules.

Gap

A finite sequence table cannot establish the universal tail. The required bridge identifies both every possible output word and a complete legal shuffle producing each parameterized word. The source and bounded literature scope are recorded in Library/Digit/layman2011a193020.md. That inspected scope does not certify worldwide openness or priority.

Route

For k at least three, set m=k-2. The ordinary binary expansion of 2^k-4 consists of m ones followed by two zeros. For every natural m, including zero, the distinct self-shuffle support of this word consists exactly of

$$1^{m+(m-i)}01^{j}01^{i-j}00,0\le j\le i\le m.$$

Exhaustion describes the three runs of ones by lengths a,b,c with a+b+c=2m and a at least m; put i=b+c and j=b. Conversely, take all m ones from the first copy, m-i ones from the second, the first zero from the first copy, j ones from the second, the second zero from the first copy, then the remaining i-j ones and two zeros from the second. This preserves both source orders and consumes both words completely.

The unary-gap uniqueness theorem recovers i and j from the output word, including when a gap is empty. The triangular parameter set has cardinality

$$\sum _{i=0}^m(i+1)=\frac{(m+2)(m+1)}{2}.$$

Substituting m=k-2 gives k(k-1)/2 for every k at least three. Subtraction and division in the Lean statement are natural-number operations; the selected range supplies the stated positive numeral.

Falsifier

A counterexample to the eventual assertion would require, for every natural K at least three, some k at least K with a different count. A single failure at k at least three would refute the stronger uniform threshold-three conclusion, but would not by itself refute eventuality. A finite matching prefix proves neither conclusion.

Evidence

D5/S1/Words/DoubleTailShuffleSupport.result has statement identity sha256:f7d5a4314bc9231c36f1523e367338917bb5974d3b87d60343ad12017a198595. It retains the existential threshold, the lower bound 3 and the universal tail without additional mathematical hypotheses; its proof chooses K=3. The accepted axiom closure is propext, Classical.choice and Quot.sound.

D5/S1/Words/DoubleTailShuffleSupport.self_shuffle_card gives the exact support cardinality for every natural m. The theorem D5/S1/Words/DoubleTailShuffleSupport.binary_root_original supplies the ordinary binary numeral bridge for every k at least three. The operation ordinaryShuffles consumes both operands in order, and conversion to a finite set precedes cardinality. For positive inputs, binaryWord is the ordinary most-significant-first numeral without leading zeros. Its empty-list totalization at zero is unused by this selected tail.

Triage

Proved: the triangular parameterization gives both exhaustive support and an injective construction of all outputs. Together with the ordinary binary numeral bridge, it proves this exact unbounded minus-four family and the whole eventual assertion with K=3.

The plus-one assertion has its own existing theorem and dossier. Other A193020 assertions remain separate questions; this resolution claim supplies no automatic conclusion about them.

ASSUMED-UNVERIFIED

The external source identity, revision-13 interpretation and bounded literature limits are those recorded in the library note. No fresh external search is asserted here. Restivo’s full chapter, the Li-Qin and Guo-Xie journal bodies, uninspected Datko-Pawlik tables and displayed lists, remaining citation branches, and broader or unindexed work were not read. Later revisions, worldwide openness and first-publication priority are not certified.

The count definition and eventual assertion are credited to John W. Layman and the OEIS contributors under CC-BY-SA-4.0. The support proof is from this repository, with source acknowledgement. Only this separate minus-four question is attached to the theorem’s typed resolution claim.