Geodesic
Restricted permutations, continued fractions, and Chebyshev polynomials
The electronic journal of combinatorics, Tome 7 (2000)
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Let $f_n^r(k)$ be the number of 132-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $12\dots k$, and let $F_r(x;k)$ and $F(x,y;k)$ be the generating functions defined by $F_r(x;k)=\sum_{n\ge 0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\ge 0}F_r(x;k)y^r$. We find an explicit expression for $F(x,y;k)$ in the form of a continued fraction. This allows us to express $F_r(x;k)$ for $1\le r\le k$ via Chebyshev polynomials of the second kind.
DOI : 10.37236/1495
Classification : 05A05, 05A15, 42C05, 30B70
Mots-clés : permutations, generating functions, continued fraction, Chebyshev polynomials 
@article{10_37236_1495,
     author = {Toufik Mansour and Alek Vainshtein},
     title = {Restricted permutations, continued fractions, and {Chebyshev} polynomials},
     journal = {The electronic journal of combinatorics},
     year = {2000},
     volume = {7},
     doi = {10.37236/1495},
     zbl = {0940.05001},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1495/}
}
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%A Toufik Mansour
%A Alek Vainshtein
%T Restricted permutations, continued fractions, and Chebyshev polynomials
%J The electronic journal of combinatorics
%D 2000
%V 7
%U http://geodesic.mathdoc.fr/articles/10.37236/1495/
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Toufik Mansour; Alek Vainshtein. Restricted permutations, continued fractions, and Chebyshev polynomials. The electronic journal of combinatorics, Tome 7 (2000). doi: 10.37236/1495

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