Provisional census of all n=6, L=872 words
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Vlad Gheorghe
Aug 11, 2026, 4:35:12 PM (2 days ago) Aug 11
to Superpermutators
Hi all, after proving that a(6)=872, I wanted to apply the findings to getting a full census.
I was waiting to finalize the proof, but given the current discussion, I am releasing provisional results. GPT believes there is 92% chance that this is the full census.
The current reconstructed catalog contains:
- 45,116 canonical words when symbol relabelling is identified and reversal is kept distinct;
- an exact expansion to 32,483,520 labelled words on the fixed alphabet 123456;
- 22,560 classes if normalized reversal is also identified;
- 40 canonical words with repeated permutation-window occurrences, forming 20 normalized-reversal pairs.
Every listed word has length 872, starts with 123456, uses exactly the six symbols, and contains all 720 distinct permutations as consecutive length-six windows. The corpus is sorted and duplicate-free.
Canonical and labelled forms
For any labelled word, read its first six symbols as p1p2p3p4p5p6. Replace p1→1, p2→2, ..., p6→6. The result is the unique canonical word, which starts with 123456.
To recover a labelled form, choose any permutation p1p2p3p4p5p6 and apply the inverse replacement 1→p1, 2→p2, ..., 6→p6 to the complete canonical word. Each canonical word therefore has exactly 720 labelled forms.
The release gives reproducible provisional IDs:
- canonical rows: A6-872-P1-C00001 through A6-872-P1-C45116, in bytewise lexicographic order;
- relabellings: A6-872-P1-R001 through A6-872-P1-R720, ordered by the first permutation;
- labelled words: A6-872-P1-L00000001 through A6-872-P1-L32483520.
For zero-based ranks, labelled_rank = 720 × canonical_rank + relabelling_rank. The package includes a small script that reconstructs the complete 872-symbol word from any labelled ID. Reversal is already represented by another canonical row, so it does not add another factor of two.
The 40 words with repeated permutation windows
A permutation-window occurrence is a length-six window that contains each symbol once. Every word in this catalog covers all 720 distinct permutations. Most words contain exactly 720 such windows, but 40 words contain an extra occurrence:
- 45,076 words contain 720 permutation-window occurrences;
- 36 words contain 721 occurrences, because one permutation occurs at two positions;
- four words contain 722 occurrences, because two distinct permutations each occur at two positions.
The 40 words form 20 normalized-reversal pairs. The attached 40-row table gives the P1 ID, the repeated permutation or permutations, every zero-based start position, the normalized-reversal partner, and the matching source filename when applicable.
This can happen without making the word longer. A permutation can occur inside an overlap bridge before the selected route reaches the vertex for that same permutation. The extra occurrence changes the number of valid windows, but it does not change the length or the set of 720 distinct permutations covered.
I also compared @Tomaz Kristan's six attached strings with this catalog. All six are present verbatim. In attachment order, their full P1 IDs are A6-872-P1-C22407, A6-872-P1-C22410, A6-872-P1-C22412, A6-872-P1-C22415, A6-872-P1-C22408, and A6-872-P1-C22413. The first four have 721 occurrences and the last two have 722. They represent six of the 20 reversal pairs; their six normalized reversals are also present in the 40-row table. The last two posted strings and their reversals account for all four 722-occurrence rows.
Status and files
The attached package contains the full 45,116-row canonical TSV, the 720-row relabelling table, a 40-row repeated-window table, a human-readable workbook, a manifest, hashes, and the labelled-word expansion script.
This is a strong word-level and reconstruction result, but it is not yet a completed census theorem.
Independent checks, comparisons, and attempts to find a valid length-872 word outside this catalog are very welcome.
I was waiting to finalize the proof, but given the current discussion, I am releasing provisional results. GPT believes there is 92% chance that this is the full census.
The current reconstructed catalog contains:
- 45,116 canonical words when symbol relabelling is identified and reversal is kept distinct;
- an exact expansion to 32,483,520 labelled words on the fixed alphabet 123456;
- 22,560 classes if normalized reversal is also identified;
- 40 canonical words with repeated permutation-window occurrences, forming 20 normalized-reversal pairs.
Every listed word has length 872, starts with 123456, uses exactly the six symbols, and contains all 720 distinct permutations as consecutive length-six windows. The corpus is sorted and duplicate-free.
Canonical and labelled forms
For any labelled word, read its first six symbols as p1p2p3p4p5p6. Replace p1→1, p2→2, ..., p6→6. The result is the unique canonical word, which starts with 123456.
To recover a labelled form, choose any permutation p1p2p3p4p5p6 and apply the inverse replacement 1→p1, 2→p2, ..., 6→p6 to the complete canonical word. Each canonical word therefore has exactly 720 labelled forms.
The release gives reproducible provisional IDs:
- canonical rows: A6-872-P1-C00001 through A6-872-P1-C45116, in bytewise lexicographic order;
- relabellings: A6-872-P1-R001 through A6-872-P1-R720, ordered by the first permutation;
- labelled words: A6-872-P1-L00000001 through A6-872-P1-L32483520.
For zero-based ranks, labelled_rank = 720 × canonical_rank + relabelling_rank. The package includes a small script that reconstructs the complete 872-symbol word from any labelled ID. Reversal is already represented by another canonical row, so it does not add another factor of two.
The 40 words with repeated permutation windows
A permutation-window occurrence is a length-six window that contains each symbol once. Every word in this catalog covers all 720 distinct permutations. Most words contain exactly 720 such windows, but 40 words contain an extra occurrence:
- 45,076 words contain 720 permutation-window occurrences;
- 36 words contain 721 occurrences, because one permutation occurs at two positions;
- four words contain 722 occurrences, because two distinct permutations each occur at two positions.
The 40 words form 20 normalized-reversal pairs. The attached 40-row table gives the P1 ID, the repeated permutation or permutations, every zero-based start position, the normalized-reversal partner, and the matching source filename when applicable.
This can happen without making the word longer. A permutation can occur inside an overlap bridge before the selected route reaches the vertex for that same permutation. The extra occurrence changes the number of valid windows, but it does not change the length or the set of 720 distinct permutations covered.
I also compared @Tomaz Kristan's six attached strings with this catalog. All six are present verbatim. In attachment order, their full P1 IDs are A6-872-P1-C22407, A6-872-P1-C22410, A6-872-P1-C22412, A6-872-P1-C22415, A6-872-P1-C22408, and A6-872-P1-C22413. The first four have 721 occurrences and the last two have 722. They represent six of the 20 reversal pairs; their six normalized reversals are also present in the 40-row table. The last two posted strings and their reversals account for all four 722-occurrence rows.
Status and files
The attached package contains the full 45,116-row canonical TSV, the 720-row relabelling table, a 40-row repeated-window table, a human-readable workbook, a manifest, hashes, and the labelled-word expansion script.
This is a strong word-level and reconstruction result, but it is not yet a completed census theorem.
Independent checks, comparisons, and attempts to find a valid length-872 word outside this catalog are very welcome.
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