Abstract
Two combinatorial identities are considered. These identities imply that \(\exp \) and \(\log \) are mutually inverse to one another even in the case where the variable is a noncommutative symbol. The proofs are given by enumeration of permutations and flags in a finite set.
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References
Konno, N., Mitsuhashi, H., Sato, I.: The quaternionic weighted zeta function of a graph. J. Alg. Comb. 44, 729–755 (2016)
Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford University Press, Oxford (1979)
Morita, H.: Ruelle zeta functions for finite digraphs, preprint
Acknowledgements
The authors would like to thank Professor Norio Konno, Yokohama National University, for illuminating discussions and valuable comments. The authors also wish to thank Professor Motohiro Ishii, Gunma University, for his many helpful comments on the proof of Theorem 1.
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Ishikawa, A., Morita, H. A combinatorial proof for the relation between the exponential function and the logarithmic function. Aequat. Math. 94, 345–354 (2020). https://doi.org/10.1007/s00010-020-00709-2
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DOI: https://doi.org/10.1007/s00010-020-00709-2