Wiseman A388713 maximum-refined partition identity

OEIS A388713 asks whether the number of superdiagonal reversed partitions of $n$ with maximum $k$ equals the number of integer partitions of $n$ with maximum $k$ and choosable initial intervals. The conjecture is preregistered in #18603.

Statement echo

The formal count T n k filters Nat.Partition n by p.parts.sup = k and superdiagonality of the ascending sort. The comparison count choosable n k uses the same maximum condition and choosability of the descending sort. The source predicates and sorting convention are imported from D5/S1/Words/Compositions/ChoosableInitialIntervals.

This is an externally stated OEIS conjecture. The delivered result is D5/S1/Words/Compositions/WisemanA388713.result, with admission basis open-problem-resolution (#18603; Proved).

The imported pointwise equivalence identifies the two predicates on every multiset. Conjoining the common maximum condition and taking finite cardinalities supplies the maximum-refined identity.

For every partition, the reverse of WisemanA388712.sorted_choosable_iff_superdiagonal gives superdiagonality of the ascending parts exactly when the descending parts are choosable. The shared predicate p.parts.sup = k is preserved, so the two filters defining T and choosable are equal pointwise. WisemanA388713.result applies finite-cardinality congruence.

The empty partition has maximum zero and both formal counts are one at $(0,0)$; the printed source row uses zero there. The identity concerns the common formal convention.

A multiset whose sorted predicates disagree, or a pair $(n,k)$ with unequal filtered cardinalities, would refute the statement. The theorem has no finite cutoff.

The public result is kernel-checked in D5/S1/Words/Compositions/WisemanA388713.lean and mirrored by its Scribe. Its proof uses the imported WisemanA388712.sorted_choosable_iff_superdiagonal helper. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

The four-slot escape audit remains unfinished for the public result; #22068 records the missing typed infinite-family registration evidence. No declared_validated registration is claimed.

The finite comparison uses the pinned experiment directory. The command python3 /Users/auric/trureturing-experiments/docs/reports/oeis-a388712-a387118-a388713-choosable-initial-intervals/check.py 22 exits $0$; check.py has SHA-256 2d3874df9fddbc803aa9b2465993e883039fff3da1b51691245f6297ffa335c1. It exhausts positive partitions for $1 \le n \le 22$. The published-prefix comparison for this entry has scope $n \le 11$.

1. Proved, kernel-checked: the equality T n k = choosable n k for every pair of natural numbers, by D5/S1/Words/Compositions/WisemanA388713.result.
2. Proved, kernel-checked: the shared pointwise predicate equivalence, supplied by D5/S1/Words/Compositions/WisemanA388712.sorted_choosable_iff_superdiagonal.
3. Computed: the pinned experiment exhausts positive partitions through $n=22$ and compares the published A388713 prefix through $n=11$; python3 check.py 22 exits 0 with script SHA-256 2d3874df9fddbc803aa9b2465993e883039fff3da1b51691245f6297ffa335c1.
4. Open: literature completeness and publication priority beyond the recorded search scope.

The finite computation has only the stated range and uses the source row-zero convention for comparison. The literature screen is bounded and does not establish a priority claim. The escape audit is unfinished at #22068.