### Pfaffian formulas for boundary connections in trees and dimers

```The double-dimer model
and skew Young diagrams
Richard W. Kenyon
David B. Wilson
Brown University
Microsoft Research
The double-dimer model
Kasteleyn matrix
1
1
3
2
2
3
5
5
4
4
6
6
8
9
9
7
7
8
(bipartite version)
The double-dimer model
4
5
5
5
3
6
6
1
4
4
2
6
1
3
3
2
2
1
The double-dimer model
4
5
3
2
6
1
The double-dimer model
Assume (wlog) that nodes alternate in color
4
5
3
2
6
1
Probability of pairing in DD model
4
5
3
2
6
1
5
4
5
3
6
1
2
4
5
3
6
1
2
4
5
3
6
1
2
4
5
3
6
1
2
4
3
6
1
2
Other connection topologies
5
4
5
3
6
1
2
5
4
4
5
3
6
1
2
4
5
3
6
1
2
4
5
3
6
1
4
3
6
2
1
2
3
6
1
5
2
4
5
3
6
1
2
4
3
6
1
2
For six nodes, get all 5 connection topologies
What if there are more nodes?
With 2n nodes, # crossing types is nth Catalan number.
Can we compute all their probabilities?
q0
q2
q3
q1
q2
q4
q1
q3
q4
q2
q3
q5
PSfrag replacement s
q6
1 £ (1 + q) £ (1 + q + q2 ) £ (1 + q + q2 + q3 )
) 1+ 2q+ 3q + 3q + 2q + q =
1 £ (1 + q) £ 1 £ 1
2
3
4
5
q5
q4
q2
q2
q1
q3
q4
q3
q2
q0
q3
q2
q3
q2
q1
q4
q3
q1
acement s
q6
) 1 + 3q + 5q2 + 5q3 + 3q4 + q5 = 1 £ (1 + q) £ (1 + q + q2 ) £ (1 + q + q2 )
```