[#P101198] Permutations whose index sums are always triangular numbers

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 \(n \ge 5\)?

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Definitions and notation

1Problem setup

Definition 1. A triangular number has the form k(k+1)/2.

Remark 1. Equivalently, this asks for a perfect matching in the bipartite graph joining i to j whenever i+j is triangular.

2What 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.

1Status

Saved packet · July 24, 2026

What counts as a solution

Saved solution (Orders 1 and 4 are the exact exceptions). A triangular-sum permutation exists for every positive order except \(1\) and \(4\).^[2]

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.

1Packet records

Show recorded summaries

Record	Kind	Assessment
A triangular-sum permutation exists for every positive order except \(1\) and \(4\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 A triangular-sum permutation exists for every positive order except \(1\) and \(4\).^[2] Relevance to this problem This records a result, observation, or conjecture used to assess the target problem. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope every positive integer n Sources Reference 2 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. Open full record →
Every order admits an involution whose termwise sums are either 2 or triangular.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 2 Every order admits an involution whose termwise sums are either 2 or triangular.^[1] Relevance to this problem This records a result, observation, or conjecture used to assess the target problem. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope every positive integer n for Sun's variant allowing sums 2 or triangular Sources Reference 1 Argument Sun's Corollary 1.2 gives an involution \(\sigma\) on every \([n]\) such that each \(k+\sigma(k)\) belongs to \(\{2\}\cup\{T_j:j\geq2\}\). Remark 1.2 identifies order 4 as a case where the sum 2 is needed. This result supplies the reflection method and explains the exceptional behavior, though it does not state the candidate's exact one-based classification. Related: Proposition 1 Open full record →
Exact bipartite matching finds failures only at orders \(1\) and \(4\) through \(n=260\).	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Exact bipartite matching finds failures only at orders \(1\) and \(4\) through \(n=260\). Relevance to this problem This supplies a computation, dataset, or executable check used to assess the target problem. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every positive integer n from 1 through 260 File research/candidate-screening/verify.py Details For each order, the checker joins left vertex \(i\) to right vertex \(j\) exactly when \(i+j\) is triangular, then searches for a perfect matching. It records failures at 1 and 4 and succeeds at every other order through 260. Related: Proposition 1 Open full record →
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Notes and companion material

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 1. For n = 4 no such permutation exists.

Recorded example 2. For n = 5 one exists.

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.

How the 3 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemPermutations whose index sums are always triangular numbers

All 2 recorded relations between these records and the problem

2See also

Density of losing positions in the subtract-a-triangular gamecombinatorics
Density of losing positions in the subtract-a-cube gamecombinatorics
Determinants of the disjoint-binary-support matrixcombinatorics

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“Permutations whose index sums are always triangular numbers.” TheoremDB. P101198. Problem statement; statement text SHA-256 240f971e12ba25d03f2398d22f2c0864c076c29568016e87874a686ccc14b36c. https://theoremdb.org/statement/?ref=P101198

BibTeX

@misc{theoremdb-problem-240f971e12ba25d03f2398d22f2c0864c076c29568016e87874a686ccc14b36c,
  title = {{Permutations whose index sums are always triangular numbers}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 240f971e12ba25d03f2398d22f2c0864c076c29568016e87874a686ccc14b36c},
  url = {https://theoremdb.org/statement/?ref=P101198}
}

Plain text: Built Markdown snapshot

This problem includes 3 records joined by 2 typed links, sourced from doi.org^[1], current as of July 24, 2026.

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1References

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

Packet source. Zhi-Wei Sun, Journal of Algebraic Combinatorics 54 (2021), Corollary 1.2 and Remark 1.2, page 895. ↗scholarly publication · reference sourceSource metadata incomplete: publication-style citation.Source use: citation only.Source named by the research packet.
Zhi-Wei Sun, On permutations of {1,...,n} and related topics, Theorem 1.1, pages 894-897; strengthened argument recorded here. ↗preprint · reference source · checked 2026-07-24Source metadata incomplete: publication-style citation, source version, relevance note.Source use: citation only.

Original question generated by an agent and settled computationally for n up to 260.

[#P101198] Permutations whose index sums are always triangular numbers

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