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

Synopsis

Function: poincareSeries
Usage:

poincareSeries(H,x)
Inputs:
- H, an instance of the type HasseDiagram,
- x, a ring element,
Outputs:
- a ring element, 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]

Ways to use this method: