Exploring OEIS A359012: The Permutation Substring Sequence

A Strange Permutation Substring Sequence: Introducing OEIS A359012

John Samuel has introduced integer sequence A359012 to the OEIS, documenting numbers formed by concatenating digits (x) and (y) that appear within the decimal expansion of xPy. Brute-force generation reveals that 708 out of the first 712 witnesses for these terms end in zero.

Why This Matters

While theoretical combinatorics often explores the properties of permutations in isolation, A359012 highlights the intersection of base-10 representation and factorial-based functions. This study reveals significant structural biases in empirical number theory, such as the fact that 565 out of 712 terms are concentrated at 6 digits, suggesting that decimal embedding is not uniformly distributed across the permutation space.

Key Insights

• 712 terms identified below 997,398 using a brute-force generator (Samuel, 2026).
• High concentration of term lengths at 6 digits, accounting for 565 of the 712 discovered terms.
• Significant trailing digit bias where 99.4% of witness values end in zero.
• Extreme sparsity in sequence distribution with a maximum gap between consecutive terms of 26,763.
• Data-driven workflow using Python for a generator-to-CSV-to-analysis pipeline.

Practical Applications

• System: Sequence Atlas dashboard. Use case: Visualizing mathematical datasets directly from CSV files for reproducible research. Pitfall: Brute-force generation for large k-values requires significant compute as xPy values grow factorially.
• System: OEIS Submission Pipeline. Use case: Validating new integer sequences through automated analysis scripts. Pitfall: Empirical bias in small datasets may obscure the rarity of features like palindromic terms.

References:

• https://oeis.org/A359012
• https://johnsamuel.info/A359012/
• https://github.com/johnsamuelwrites/A359012