# poincareSeries(HasseDiagram,RingElement) -- the generating series of a Hasse diagram

## Synopsis

• Function: poincareSeries
• Usage:
poincareSeries(H,x)
• Inputs:
• Outputs:
• , a polynomial in the variable x where the coefficient in front of x^i is the number of elements in the i-th row of H (counting from 0).

## Description

 i1 : R=rootSystemA(2) o1 = RootSystem{...8...} o1 : RootSystem i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R) o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, | 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}} | -1 | | 2 | | -1 | | -2 | | -1 | | 1 | | 1 | | 1 | | 2 | | -1 | | 2 | | 2 | | -1 | | 1 | o2 : HasseDiagram i3 : ZZ[x] o3 = ZZ[x] o3 : PolynomialRing i4 : poincareSeries(H,x) 3 2 o4 = x + 2x + 2x + 1 o4 : ZZ[x]