An explicit bijection between the inversion sequences avoiding 0312 and 0321

Year 2026, Volume: 55 Issue: 1, 53 - 59, 23.02.2026

Melis Gezer , Gökhan Yıldırım

https://doi.org/10.15672/hujms.1602878

https://izlik.org/JA52DX46ZE

Abstract

We present an explicit bijection between the set of inversion sequences avoiding the patterns 0312 and 0321, preserving five statistics: the counts of zeros, distinct elements, repeating elements, left-to-right maxima, and the maximum entry. We also provide some sampling algorithms for these pattern-avoiding inversion sequences

Keywords

inversion sequences , pattern avoidance , random sampling

Supporting Institution

Tübitak

Project Number

Tübitak-Ardeb grant no: 120F352

References

• [1] J. S. Auli and S. Elizalde, Consecutive patterns in inversion sequences, Discrete Math. Theor. Comput. Sci. 21 (2), Paper No. 6, 22 pp, 2019.
• [2] M. Bona, Combinatorics of Permutations. Chapman Hall/CRC, Boca Raton, FL, 2. edition edition, 2004.
• [3] S. Chern, On 0012-avoiding inversion sequences and a conjecture of Lin and Ma., Quaest. Math. 46 (4), 681–694, 2023.
• [4] S. Corteel, M. A. Martinez, C. D. Savage, and M. Weselcouch, Patterns in inversion sequences I, Discrete Math. Theor. Comput. Sci. 18 (2), 2015.
• [5] L. Hong and R. Li, Length-four pattern avoidance in inversion sequences, Electron. J. Combinatorics, 29 (4), 2022.
• [6] S. Kitaev, Patterns in Permutations and Words, Monographs in Theoretical Computer Science. Springer, 2011.
• [7] I. Kotsireas, T. Mansour, and G. Yıldırım, An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences, J. Symbolic Comput. 120, Paper No. 102231, 2024.
• [8] S. Liu, Statistics on trapezoidal words and k-inversion sequences, Discrete Appl. Math. 325, 1–8, 2023.
• [9] S. Liu, r-Euler-Mahonian statistics on permutations, J. Combin. Theory Ser. A, 208, Paper No. 105940, 30, 2024.
• [10] N. Madras and H. Liu, Random pattern-avoiding permutations, Algorithmic Probability and Combinatorics, volume 520 of Contemporary Mathematics, pages 173–194. American Mathematical Society, Providence, RI, 2010.
• [11] T. Mansour, Five classes of pattern avoiding inversion sequences under one roof: generating trees, J. Difference Equ. Appl. 29 (7), 748–762, 2023.
• [12] T. Mansour, Generating trees for 0021-avoiding inversion sequences and a conjecture of Hong and Li, Discrete Math. Lett. 12, 11–14, 2023.
• [13] T. Mansour and M. Shattuck, Pattern avoidance in inversion sequences, Pure Math. Appl. 25 (2),157–176, 2016.
• [14] T. Mansour and G. Yıldırım, Inversion sequences avoiding 021 and another pattern of length four, Discrete Math. Theor. Comput. Sci. 25 (2), 2023.
• [15] J. Pantone, The enumeration of inversion sequences avoiding the patterns 201 and 210, Enumer. Comb. Appl. 4 (4), Paper No. S2R25, 12, 2024.
• [16] L. Pudwell, From permutation patterns to the periodic table, Notices Amer. Math. Soc. 67 (7), 994–1001, 2020.
• [17] B. Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv preprint arXiv:2407.07701, 2024.
• [18] C. Yan and Z. Lin, Inversion sequences avoiding pairs of patterns, Discrete Math. Theor. Comput. Sci, 22 (1), 2020.