bibkey: oeis2025wisemanpartitions
authors: Gus Wiseman
year: 2025
title: OEIS A388712, A387118 and A388713
doi: null
url: https://oeis.org/A388712
claim: The three conjectures equate choosable initial-interval partition counts and superdiagonal reversed-partition counts, with complementation or a common statistic.
strata_touched:
  - D5/S1/Words/Compositions/WisemanA388712
  - D5/S1/Words/Compositions/WisemanA387118
  - D5/S1/Words/Compositions/WisemanA388713
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A388712

Related entries: https://oeis.org/A387118 and https://oeis.org/A388713. These OEIS entries are unpaginated; the locators below specify their Name and Comments fields. The source snapshot is oeis/oeisdata at e7d0073e.

Wiseman’s partition-count conjectures

A388712

Name:

Triangle read by rows where T(n,k) is the number of integer partitions of n into k parts without choosable initial intervals.

Comments:

Conjecture: Also the number of non-superdiagonal reversed partitions of n into k parts, where a sequence (y_1, …, y_k) is superdiagonal iff y_i >= i for all i = 1..k.

A387118

Name:

Number of integer partitions of n without choosable initial intervals.

Comments:

Conjecture: Also the number of non-superdiagonal reversed integer partitions of n. - Gus Wiseman, Oct 04 2025

A388713

Name:

Triangle read by rows where T(n,k) is the number of superdiagonal multisets summing to n (reversed partitions of n) with maximum k.

Comments:

Conjecture: Also the number of integer partitions of n with maximum k and choosable initial intervals.

A388712 also defines the objects as follows:

We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. … The initial interval of a nonnegative integer x is the set {1,…,x}.

A388713 states:

A sequence (y_1, …, y_k) is superdiagonal iff y_i >= i for all i = 1..k.

Ordinary parts are sorted decreasingly, reversed parts increasingly. The existing ChoosableInitial and Superdiagonal predicates express distinct positive representatives and the one-based bound. Nat.Partition has positive parts. The first two identities count failures of these predicates, refined by number of parts in A388712. A388713 counts successes with maximum Multiset.sup. Its empty partition has maximum zero; both Lean counts are 1 at (0,0), whereas the source prints 0 in row 0. The common empty-partition value does not affect the identity.

The proofs reuse permutation invariance and the increasing-list criterion in D5/S1/Words/Compositions/ChoosableInitialIntervals. No claim of novelty for that elementary consequence of the distinct-representative criterion is made.

Finite comparison

The experiment is docs/reports/oeis-a388712-a387118-a388713-choosable-initial-intervals/ in https://github.com/the-omega-institute/trureturing-experiments, commit 29dcaf8a7cfd8f069c88eff27acd8735cfdfc4d1. The checker check.py has sha256 2d3874df9fddbc803aa9b2465993e883039fff3da1b51691245f6297ffa335c1. The source records python3 check.py 22, exit 0, ending in RESULT PASS. It exhausts partitions through n=22 and compares all three data sets, using the source’s printed row-zero value for A388713. This finite comparison is separate from the universal Lean identities.

Literature boundary

The source screen on issues #18601, #18602 and #18603 checked the OEIS entries, repository declarations and public Lean problem collections. Related literature is P. Hall, On Representatives of Subsets, Journal of the London Mathematical Society 10 (1935), 26–30, DOI 10.1112/jlms/s1-10.37.26; Peter J. Cameron, Hall’s marriage theorem, arXiv:2503.23159v1; and M. Archibald, A. Blecher, S. Elizalde and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afrika Matematika 36, article 77 (2025), DOI 10.1007/s13370-025-01282-0. The first two state the distinct-representative criterion; they are not on these modules’ Lean dependency paths. The third studies superdiagonal partition enumeration; it is not on these modules’ dependency paths either. No prior settlement of the three named identities was found in the OEIS text, visible history and exact-name / choosability / superdiagonal searches over arXiv and web sources. This is a searched-scope statement, not a priority claim. Exhaustive literature completeness and publication priority are ASSUMED-UNVERIFIED.