It contains lots of numbers, or, rather, polynomials in q,
that give association scheme parameters v, k_{i},
p^{i}_{jk} for association schemes defined on
the geometry G/P of cosets of a parabolic subgroup P of a
finite group of Lie type G defined over the field GF(q),
especially for maximal parabolics P.

Note: the numbering of diagram nodes is non-standard here.

Look at a random valency (k_{3}in E_{7,7}): q^{11}+ 2q^{12}+ 4q^{13}+ 7q^{14}+ 10q^{15}+ 13q^{16}+ 16q^{17}+ 17q^{18}+ 17q^{19}+ 16q^{20}+ 13q^{21}+ 10q^{22}+ 7q^{23}+ 4q^{24}+ 2q^{25}+ q^{26}. What properties can one observe? The sequence of coefficients is symmetric (palindromic) and unimodal. The polynomial factors as q^{11}(q^{7}−1)(q^{5}−1)(q^{4}−1)(q^{3}+1) / (q^{2}−1)(q−1)^{2}.

Given these polynomials, one can ask all kinds of questions. Are they symmetric (palindromic)? Are they unimodal? ...

Such questions have been asked e.g. by R. Stanley, seeWeyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Alg. Discr. Meth.1(1980) 168-184, and alsoLog-concave and unimodal sequences in algebra, combinatorics, and geometry, pp. 500-535 in Graph Theory and Its Applications: East and West, Ann. New York Acad. Sci. 576, 1989. E.g., Theorem 19 in this last paper says that the number v of vertices of G/P is symmetric and unimodal (as polynomial in q).

**Exercise**
Show that all valencies k_{i} are symmetric and unimodal
(as polynomial in q). Find their factorization.

Symmetry is easy. Unimodality follows from Stanley. The factorization is found from the degrees of a suitable Coxeter subgroup.

A very conspicuous property is that the valency for the oppositeness relation is a power of q. More generally one can prove that all eigenvalues of the oppositeness graph have squares that are powers of q.

(A. E. Brouwer,The eigenvalues of oppositeness graphs in buildings of spherical type, pp. 1-10 in: Combinatorics and Graphs, R. A. Brualdi, S. Hedayat, H. Kharaghani, G. B. Khosrovshahi, S. Shahriari (eds.), AMS Contemporary Mathematics Series 531, 2010.)

What about the parameters p^{i}_{jk}?

Look at two random values in E_{7,7}. We have p^{8}_{17}= q^{2}+ 2q^{3}+ 4q^{4}+ 4q^{5}+ 4q^{6}+ 2q^{7}+ q^{8}, and p^{8}_{18}= −1 − q^{2}− q^{3}− q^{4}+ 3q^{7}+ 3q^{8}+ 4q^{9}+ 3q^{10}+ 2q^{11}+ q^{12}. We see that p^{i}_{1j}behaves well for j ≠ i, but not for j = i. Since the k_{i}factor nicely, and k_{i}p^{i}_{1j}= k_{j}p^{j}_{1i}, one expects a nice factorization for p^{i}_{1j}for j ≠ i. Also, the three-term expression for p^{i}_{1j}becomes a two-term one, with positive terms only, if j ≠ i, while p^{i}_{1i}also has contributions (q−1)q^{a}.

**Problem**
Conjecture and prove properties of p^{i}_{jk}.

A2,1 | |||

A3,1 | A3,2 | ||

A4,1 | A4,2 | ||

A5,1 | A5,2 | A5,3 | |

A6,1 | A6,2 | A6,3 | |

A7,1 | A7,2 | A7,3 | A7,4 |

B2,1 | ||||||

B3,1 | B3,2 | B3,3 | ||||

B4,1 | B4,2 | B4,3 | B4,4 | |||

B5,1 | B5,2 | B5,3 | B5,4 | B5,5 | ||

B6,1 | B6,2 | B6,3 | B6,4 | B6,5 | B6,6 | |

B7,1 | B7,2 | B7,3 | B7,4 | B7,5 | B7,6 | B7,7 |

D4,1 | D4,2 | ||||

D5,1 | D5,2 | D5,3 | D5,4 | ||

D6,1 | D6,2 | D6,3 | D6,4 | D6,5 | |

D7,1 | D7,2 | D7,3 | D7,4 | D7,5 | D7,6 |

E6,1 | E6,2 | E6,3 | E6,6 |

E7,1 | E7,6 | E7,7 | |

E8,1 | E8,7 | E8,8 | |

F4,1 | F4,2 | ||

G2,1 |

^{3}D4,1 |
^{3}D4,2 | ||

^{2}E6,1 |
^{2}E6,2 |
^{2}E6,3 |
^{2}E6,4 |

**Explanation**
Consider G and P with thin analogs W and X, so that G = BWB and P = BXB.
Let R be the set of fundamental reflections, so that W = <R>.
For r ∈ R we consider X = <R\{r}>, so that P will be
a maximal parabolic.
We give the number of cosets |W/X| and the number of double cosets |X\W/X|.
Next, for each double coset XwX an index i (in 0..|X\W/X|−1), its shortest
representative w (as a string between parentheses, the digits j indicate
the fundamental reflections r_{j}),
its size k_{i} = |XwX/X| (as a number in square brackets)
and |PwP/P| (as a polynomial in q).
The double cosets X = X1X and XrX are numbered 0 and 1.
Next we consider the graph Γ on the cosets gP, where gP, hP are adjacent
when h^{−1}g ∈ PrP and give the double coset diagram of Γ.
For each double coset XwX (viewed as a node in the double coset diagram,
and identified with the set of vertices PwP/P) we give the number
p^{i}_{1j} of neighbours that any of its vertices
(such as wP) has in any other double coset.

For example, the graphs will be strongly regular when the number
of double cosets is 3, and then the k_{i} give
1, k, v−k−1, and the p^{i}_{1j} give
λ, μ, etc.

[BCN] A. E. Brouwer, A. M. Cohen & A. Neumaier,
*Distance-regular graphs*, Springer, 1989.