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.[1][2][3]
Definitions
For a positive integer and for an integer , the Narayana number is defined by
The number is defined as for and as for .
For a nonnegative integer , the -th Narayana polynomial is defined by
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:[4]
is the -th large
Schröder number. This is the number of plane trees having edges with leaves colored by one of two colors. The first few Schröder numbers are . (sequence A006318 in the
OEIS).[5]
For integers , let denote the number of underdiagonal paths from to in a grid having step set . Then .[6]
Recurrence relations
For , satisfies the following nonlinear recurrence relation:[6]
.
For , satisfies the following second order linear recurrence relation:[6]
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.[1][2][3]
Definitions
For a positive integer and for an integer , the Narayana number is defined by
The number is defined as for and as for .
For a nonnegative integer , the -th Narayana polynomial is defined by
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:[4]
is the -th large
Schröder number. This is the number of plane trees having edges with leaves colored by one of two colors. The first few Schröder numbers are . (sequence A006318 in the
OEIS).[5]
For integers , let denote the number of underdiagonal paths from to in a grid having step set . Then .[6]
Recurrence relations
For , satisfies the following nonlinear recurrence relation:[6]
.
For , satisfies the following second order linear recurrence relation:[6]