# American Institute of Mathematical Sciences

March  2020, 28(1): 149-156. doi: 10.3934/era.2020009

## On the existence of permutations conditioned by certain rational functions

 I.R.M.A., UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, F-67084 Strasbourg, France

Received  October 2019 Revised  February 2020 Published  March 2020

We prove several conjectures made by Z.-W. Sun on the existence of permutations conditioned by certain rational functions. Furthermore, we fully characterize all integer values of the "inverse difference" rational function. Our proofs consist of both investigation of the mathematical properties of the rational functions and brute-force attack by computer for finding special permutations.

Citation: Guo-Niu Han. On the existence of permutations conditioned by certain rational functions. Electronic Research Archive, 2020, 28 (1) : 149-156. doi: 10.3934/era.2020009
##### References:
 [1] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511801655.  Google Scholar [2] R. Sedgewick, Permutation generation methods, Comput. Surveys, 9 (1977), 137-164.  doi: 10.1145/356689.356692.  Google Scholar [3] R. P. Stanley, Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012.   Google Scholar [4] Z.-W. Sun, On permutation of $\{1, \ldots, n\}$ and related topics, preprint, arXiv: 1811.10503. Google Scholar

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##### References:
 [1] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511801655.  Google Scholar [2] R. Sedgewick, Permutation generation methods, Comput. Surveys, 9 (1977), 137-164.  doi: 10.1145/356689.356692.  Google Scholar [3] R. P. Stanley, Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012.   Google Scholar [4] Z.-W. Sun, On permutation of $\{1, \ldots, n\}$ and related topics, preprint, arXiv: 1811.10503. Google Scholar
The increasing binary tree for $\delta_{32}$
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