# P101198: Permutations whose index sums are always triangular numbers

- ID: `P101198`
- Reference: `triangular-sum-permutation-existence`
- Page: https://theoremdb.org/statements/P101198
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/triangular-sum-permutation-existence?representation=markdown
- Record maturity: Archived record

## The problem

For which \(n\) does there exist a permutation \(\sigma\) of \(\{1, \dots, n\}\) such that \(i + \sigma(i)\) is a triangular number for every \(i\)? Is the set of exceptions exactly \(\{1, 4\}\), so that such a permutation exists for every \(n \ge 5\)?

### Problem setup

- **Definition.** A triangular number has the form k(k+1)/2.
- **Remark.** Equivalently, this asks for a perfect matching in the bipartite graph joining i to j whenever i+j is triangular.

### What counts as a solution

- Prove that a suitable permutation exists for every n at least 5, or exhibit some n at least 5 for which none exists.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.

A triangular-sum permutation exists for every positive order except \(1\) and \(4\). [2](#reference-2)

**Recorded scope.** every positive integer n

[Read this exact record and its supporting evidence](/records/tsp-claim-exact-classification)


### Resolution argument

Let \(T_k=k(k+1)/2\). If a working permutation is known on \([m]\), and \(T_k=m+n+1\) with \(m<n\), keep that permutation on \([m]\) and set \(\sigma(i)=T_k-i\) on \([m+1,n]\). The latter interval maps to itself and every new sum equals \(T_k\).

Working base orders are \(0,2,3,5,10\). At order 10, one witness is \((1\,2\,4\,6\,9)(5\,10)(7\,8)\), with 3 fixed. For \(n\geq6\), take the least triangular number above \(n\) and set \(m=T_k-n-1\). This gives \(0\leq m<n\). If \(m\) is 1 or 4, the next triangular number gives remainder \(k+2\) or \(k+5\), respectively; these are earlier admissible orders, apart from the base case \(n=10\). Strong induction finishes the construction.

Order 1 fails because its sole sum is 2. At order 4, left vertices 1 and 4 both have the sole neighbor 2, so Hall's condition fails. The proof is an editorial strengthening of Sun's reflection construction and should receive independent review before the problem state changes.

## Research packet

### Saved solution and evidence

**Proposition 1 (Orders 1 and 4 are the exact exceptions).** A triangular-sum permutation exists for every positive order except \(1\) and \(4\).

Let \(T_k=k(k+1)/2\). If a working permutation is known on \([m]\), and \(T_k=m+n+1\) with \(m<n\), keep that permutation on \([m]\) and set \(\sigma(i)=T_k-i\) on \([m+1,n]\). The latter interval maps to itself and every new sum equals \(T_k\).

Working base orders are \(0,2,3,5,10\). At order 10, one witness is \((1\,2\,4\,6\,9)(5\,10)(7\,8)\), with 3 fixed. For \(n\geq6\), take the least triangular number above \(n\) and set \(m=T_k-n-1\). This gives \(0\leq m<n\). If \(m\) is 1 or 4, the next triangular number gives remainder \(k+2\) or \(k+5\), respectively; these are earlier admissible orders, apart from the base case \(n=10\). Strong induction finishes the construction.

Order 1 fails because its sole sum is 2. At order 4, left vertices 1 and 4 both have the sole neighbor 2, so Hall's condition fails. The proof is an editorial strengthening of Sun's reflection construction and should receive independent review before the problem state changes.

### Background and intake notes

Screened alongside the square and cube analogues in the same matching sweep.

- Original intake status: Status not established. No literature search was performed.
- No literature search was carried out.
- Triangular numbers are denser than squares in the relevant range, which is consistent with the much smaller exception set and suggests this version may be the easier of the two.

- Recorded example: For n = 4 no such permutation exists.
- Recorded example: For n = 5 one exists.

### Other known results

- **Proposition 2** (supported): Every order admits an involution whose termwise sums are either 2 or triangular. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (executable material): Exact bipartite matching finds failures only at orders \(1\) and \(4\) through \(n=260\).

### Computational notes

- Bipartite perfect matching decided exactly for every n from 1 to 260. A permutation exists for every n in that range except 1 and 4.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `triangular-sum-permutation-existence`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## Lean verification

A solution has been submitted and awaits independent review.

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## References

2 entries have incomplete source metadata. Each affected row names the fields that still need editorial review.

1. <a id="reference-1"></a>Zhi-Wei Sun, Journal of Algebraic Combinatorics 54 (2021), Corollary 1.2 and Remark 1.2, page 895 https://doi.org/10.1007/s10801-021-01028-8
   - scholarly_publication; reference source
   - Source metadata incomplete: publication-style citation.
   - Source use: citation_only
   - Source named by the research packet.
2. <a id="reference-2"></a>Zhi-Wei Sun, On permutations of {1,...,n} and related topics, Theorem 1.1, pages 894-897; strengthened argument recorded here https://arxiv.org/abs/1811.10503
   - preprint; reference source; checked 2026-07-24
   - Source metadata incomplete: publication-style citation, source version, relevance note.
   - Source use: citation_only