J. Diehl, L. Schmitz, 2022. Two-parameter sums signatures and corresponding quasisymmetric functions. arXiv preprint arXiv:2210.14247. [https://arxiv.org/abs/2210.14247]

The 2-parameter sums signatures is implemented in twoParameterSS.py for arbitrary dimension d. The matrix composition Hopf algebra is implemented in composition_Algebra.py for arbitrary d. picturesLoding.py is a small library for image preperation based on warping. Sample pictures are provided in the folder pictures.

The code uses a library for linear combinations by J. Diehl: [https://github.com/diehlj/linear-combination-py]

tests.py contains numeric tests for the article, including

• Example 2.5. (toy computation SS)
• Example 2.7. (toy computation quasi-shuffle)
• Example 2.9 (toy computation coproduct)
• Theorem 2.10. (Hopf algebra)
• Theorem 2.11. (Quasi-shuffle identity)
• Example 2.13. (toy computation quasi-shuffle)
• Lemma 2.15. (Zero properties)
• Theorem 2.18. (SS zero insertion invariant)
• Lemma 2.19. (warp properties)
• Lemma 2.20. (delta, sigma properties)
• Lemma 2.22. (identeties relating delta, Zero, warp, normal forms)
• Theorem 2.30 (SS(delta(.)) invariant to warping)
• Example 2.32 (toy computation diagonal concatenation in evZ)
• Lemma 2.33 (Chen's identity in evZ)
• Example 2.35 (toy computation diagonal concatenaion in evC)
• Theorem 4.5 (efficient computation of SS w.r.t. chained 1x1 compositions)
• Lemma 5.6 (delta semi-group homomorphism)
• Lemma 5.7 (sigma semi-group homomorphism)

The test file also uses pictures to compute the 2-parameter signatues with d=3.

Compare also the SAGE implementation for quasisymmetric functions [https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/ncsf_qsym/qsym.html]