Narayana polynomials are a class of polynomials whose coefficients are the Narayana numbers. The Narayana numbers and Narayana polynomials are named after the Canadian mathematician T. V. Narayana (1930–1987). They appear in several combinatorial problems.
Definitions For a positive integer n {\displaystyle n} and for an integer k ≥ 0 {\displaystyle k\geq 0} , the Narayana number N ( n , k ) {\displaystyle N(n,k)} is defined by
N ( n , k ) = 1 n ( n k ) ( n k − 1 ) . {\displaystyle N(n,k)={\frac {1}{n}}{n \choose k}{n \choose k-1}.}
The number N ( 0 , k ) {\displaystyle N(0,k)} is defined as 1 {\displaystyle 1} for k = 0 {\displaystyle k=0} and as 0 {\displaystyle 0} for k ≠ 0 {\displaystyle k\neq 0} . For a nonnegative integer n {\displaystyle n} , the n {\displaystyle n} -th Narayana polynomial N n ( z ) {\displaystyle N_{n}(z)} is defined by
N n ( z ) = ∑ k = 0 n N ( n , k ) z k . {\displaystyle N_{n}(z)=\sum _{k=0}^{n}N(n,k)z^{k}.}
The associated Narayana polynomial N n ( z ) {\displaystyle {\mathcal {N}}_{n}(z)} is defined as the reciprocal polynomial of N n ( z ) {\displaystyle N_{n}(z)} :
N n ( z ) = z n N n ( 1 z ) {\displaystyle {\mathcal {N}}_{n}(z)=z^{n}N_{n}\left({\tfrac {1}{z}}\right)} .
Examples The first few Narayana polynomials are
N 0 ( z ) = 1 {\displaystyle N_{0}(z)=1}
N 1 ( z ) = z {\displaystyle N_{1}(z)=z}
N 2 ( z ) = z 2 + z {\displaystyle N_{2}(z)=z^{2}+z}
N 3 ( z ) = z 3 + 3 z 2 + z {\displaystyle N_{3}(z)=z^{3}+3z^{2}+z}
N 4 ( z ) = z 4 + 6 z 3 + 6 z 2 + z {\displaystyle N_{4}(z)=z^{4}+6z^{3}+6z^{2}+z}
N 5 ( z ) = z 5 + 10 z 4 + 20 z 3 + 10 z 2 + z {\displaystyle N_{5}(z)=z^{5}+10z^{4}+20z^{3}+10z^{2}+z}
Properties A few of the properties of the Narayana polynomials and the associated Narayana polynomials are collected below. Further information on the properties of these polynomials are available in the references cited.
Alternative form of the Narayana polynomials The Narayana polynomials can be expressed in the following alternative form:
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