Perfect Matching of 2k
Perfect Match of 2k, with 2k at the end (back to Data page)
定義：
- Perfect Matching of [2n]:
A set partition of [2n] with blocks (disjoint nonempty subsets) of size exactly 2. Let $\mathit{M}_{2n}$ be the set of matchings of [2n], and let $M \in \mathit{M}_{2n}$. The standard form of $M$ is a list of blocks $\{(i_1, j_1), (i_2, j_2), ..., (i_n, j_n) \}$ such that $i_r < j_r$ for all $1\leq r\leq n$ and $1=i_1 < i_2 < ... < i_n$.
- Perfect Matching of [2k] with 2k at the end:
For above $M$ must have $j_k=2k$.
- $ee(\rho)$ =
#$\{i \mid \rho(i) \mbox{ is even and } \rho(i+1) \mbox{ is even}\}$
- $eo(\rho)$ =
#$\{i \mid \rho(i) \mbox{ is even and } \rho(i+1) \mbox{ is odd}\}$
- $oe(\rho)$ =
#$\{i \mid \rho(i) \mbox{ is odd and } \rho(i+1) \mbox{ is even}\}$
- $oo(\rho)$ =
#$\{i \mid \rho(i) \mbox{ is odd and } \rho(i+1) \mbox{ is odd}\}$
目錄：
總數
| k | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| total | 2 | 8 | 48 | 384 | 3840 | 46080 | 645120 |
A000165,
A241122
分項
| k |
total |
numbers | ee | eo | oe | oo |
| 2 |
2 |
1 | 0 | 0 | 2 | 0 |
| 1 | 1 | 0 | 0 | 1 |
| 3 |
8 |
1 | 0 | 0 | 3 | 0 |
| 3 | 1 | 0 | 1 | 1 |
| 2 | 0 | 1 | 2 | 0 |
| 2 | 1 | 1 | 0 | 1 |
| 4 |
48 |
1 | 0 | 0 | 4 | 0 |
| 12 | 1 | 0 | 2 | 1 |
| 7 | 0 | 1 | 3 | 0 |
| 17 | 1 | 1 | 1 | 1 |
| 5 | 2 | 0 | 0 | 2 |
| 2 | 0 | 2 | 2 | 0 |
| 4 | 1 | 2 | 0 | 1 |
| 5 |
384 |
1 | 0 | 0 | 5 | 0 |
| 45 | 1 | 0 | 3 | 1 |
| 18 | 0 | 1 | 4 | 0 |
| 123 | 1 | 1 | 2 | 1 |
| 52 | 2 | 0 | 1 | 2 |
| 25 | 0 | 2 | 3 | 0 |
| 65 | 1 | 2 | 1 | 1 |
| 45 | 2 | 1 | 0 | 2 |
| 2 | 0 | 3 | 2 | 0 |
| 8 | 1 | 3 | 0 | 1 |
| 6 |
3840 |
1 | 0 | 0 | 6 | 0 |
| 156 | 1 | 0 | 4 | 1 |
| 41 | 0 | 1 | 5 | 0 |
| 791 | 1 | 1 | 3 | 1 |
| 446 | 2 | 0 | 2 | 2 |
| 145 | 0 | 2 | 4 | 0 |
| 768 | 1 | 2 | 2 | 1 |
| 824 | 2 | 1 | 1 | 2 |
| 63 | 0 | 3 | 3 | 0 |
| 225 | 1 | 3 | 1 | 1 |
| 93 | 3 | 0 | 0 | 3 |
| 269 | 2 | 2 | 0 | 2 |
| 2 | 0 | 4 | 2 | 0 |
| 16 | 1 | 4 | 0 | 1 |
| 7 |
46080 |
1 | 0 | 0 | 7 | 0 |
| 513 | 1 | 0 | 5 | 1 |
| 88 | 0 | 1 | 6 | 0 |
| 4485 | 1 | 1 | 4 | 1 |
| 3367 | 2 | 0 | 3 | 2 |
| 638 | 0 | 2 | 5 | 0 |
| 7880 | 1 | 2 | 3 | 1 |
| 10369 | 2 | 1 | 2 | 2 |
| 774 | 0 | 3 | 4 | 0 |
| 4021 | 1 | 3 | 2 | 1 |
| 2061 | 3 | 0 | 1 | 3 |
| 7690 | 2 | 2 | 1 | 2 |
| 141 | 0 | 4 | 3 | 0 |
| 745 | 1 | 4 | 1 | 1 |
| 1908 | 3 | 1 | 0 | 3 |
| 1365 | 2 | 3 | 0 | 2 |
| 2 | 0 | 5 | 2 | 0 |
| 32 | 1 | 5 | 0 | 1 |
| 8 |
645120 |
1 | 0 | 0 | 8 | 0 |
| 1632 | 1 | 0 | 6 | 1 |
| 183 | 0 | 1 | 7 | 0 |
| 23099 | 1 | 1 | 5 | 1 |
| 22925 | 2 | 0 | 4 | 2 |
| 2440 | 0 | 2 | 6 | 0 |
| 69002 | 1 | 2 | 4 | 1 |
| 111097 | 2 | 1 | 3 | 2 |
| 6150 | 0 | 3 | 5 | 0 |
| 60212 | 1 | 3 | 3 | 1 |
| 30999 | 3 | 0 | 2 | 3 |
| 139413 | 2 | 2 | 2 | 2 |
| 3261 | 0 | 4 | 4 | 0 |
| 19438 | 1 | 4 | 2 | 1 |
| 61653 | 3 | 1 | 1 | 3 |
| 57490 | 2 | 3 | 1 | 2 |
| 299 | 0 | 5 | 3 | 0 |
| 2401 | 1 | 5 | 1 | 1 |
| 3789 | 4 | 0 | 0 | 4 |
| 23205 | 3 | 2 | 0 | 3 |
| 6365 | 2 | 4 | 0 | 2 |
| 2 | 0 | 6 | 2 | 0 |
| 64 | 1 | 6 | 0 | 1 |
Data
n=2, Total: 2
1. [(1, 2), (3, 4)] (ee=0, eo=0, oe=2, oo=0)
2. [(1, 3), (2, 4)] (ee=1, eo=0, oe=0, oo=1)
n=3, Total: 8
1. [(1, 2), (3, 4), (5, 6)] (ee=0, eo=0, oe=3, oo=0)
2. [(1, 2), (3, 5), (4, 6)] (ee=1, eo=0, oe=1, oo=1)
3. [(1, 3), (2, 4), (5, 6)] (ee=1, eo=0, oe=1, oo=1)
4. [(1, 4), (2, 3), (5, 6)] (ee=0, eo=1, oe=2, oo=0)
5. [(1, 3), (2, 5), (4, 6)] (ee=1, eo=1, oe=0, oo=1)
6. [(1, 5), (2, 3), (4, 6)] (ee=1, eo=1, oe=0, oo=1)
7. [(1, 4), (2, 5), (3, 6)] (ee=0, eo=1, oe=2, oo=0)
8. [(1, 5), (2, 4), (3, 6)] (ee=1, eo=0, oe=1, oo=1)
n=4, Total: 48
1. [(1, 2), (3, 4), (5, 6), (7, 8)] (ee=0, eo=0, oe=4, oo=0)
2. [(1, 2), (3, 4), (5, 7), (6, 8)] (ee=1, eo=0, oe=2, oo=1)
3. [(1, 2), (3, 5), (4, 6), (7, 8)] (ee=1, eo=0, oe=2, oo=1)
4. [(1, 2), (3, 6), (4, 5), (7, 8)] (ee=0, eo=1, oe=3, oo=0)
5. [(1, 2), (3, 5), (4, 7), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
6. [(1, 2), (3, 7), (4, 5), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
7. [(1, 2), (3, 6), (4, 7), (5, 8)] (ee=0, eo=1, oe=3, oo=0)
8. [(1, 2), (3, 7), (4, 6), (5, 8)] (ee=1, eo=0, oe=2, oo=1)
9. [(1, 3), (2, 4), (5, 6), (7, 8)] (ee=1, eo=0, oe=2, oo=1)
10. [(1, 4), (2, 3), (5, 6), (7, 8)] (ee=0, eo=1, oe=3, oo=0)
11. [(1, 3), (2, 4), (5, 7), (6, 8)] (ee=2, eo=0, oe=0, oo=2)
12. [(1, 4), (2, 3), (5, 7), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
13. [(1, 3), (2, 5), (4, 6), (7, 8)] (ee=1, eo=1, oe=1, oo=1)
14. [(1, 3), (2, 6), (4, 5), (7, 8)] (ee=1, eo=1, oe=1, oo=1)
15. [(1, 5), (2, 3), (4, 6), (7, 8)] (ee=1, eo=1, oe=1, oo=1)
16. [(1, 6), (2, 3), (4, 5), (7, 8)] (ee=0, eo=2, oe=2, oo=0)
17. [(1, 4), (2, 5), (3, 6), (7, 8)] (ee=0, eo=1, oe=3, oo=0)
18. [(1, 4), (2, 6), (3, 5), (7, 8)] (ee=1, eo=0, oe=2, oo=1)
19. [(1, 5), (2, 4), (3, 6), (7, 8)] (ee=1, eo=0, oe=2, oo=1)
20. [(1, 5), (2, 6), (3, 4), (7, 8)] (ee=1, eo=0, oe=2, oo=1)
21. [(1, 6), (2, 4), (3, 5), (7, 8)] (ee=1, eo=0, oe=2, oo=1)
22. [(1, 6), (2, 5), (3, 4), (7, 8)] (ee=0, eo=1, oe=3, oo=0)
23. [(1, 3), (2, 5), (4, 7), (6, 8)] (ee=1, eo=2, oe=0, oo=1)
24. [(1, 3), (2, 7), (4, 5), (6, 8)] (ee=1, eo=2, oe=0, oo=1)
25. [(1, 5), (2, 3), (4, 7), (6, 8)] (ee=1, eo=2, oe=0, oo=1)
26. [(1, 7), (2, 3), (4, 5), (6, 8)] (ee=1, eo=2, oe=0, oo=1)
27. [(1, 3), (2, 6), (4, 7), (5, 8)] (ee=1, eo=1, oe=1, oo=1)
28. [(1, 3), (2, 7), (4, 6), (5, 8)] (ee=1, eo=1, oe=1, oo=1)
29. [(1, 6), (2, 3), (4, 7), (5, 8)] (ee=0, eo=2, oe=2, oo=0)
30. [(1, 7), (2, 3), (4, 6), (5, 8)] (ee=1, eo=1, oe=1, oo=1)
31. [(1, 4), (2, 5), (3, 7), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
32. [(1, 4), (2, 7), (3, 5), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
33. [(1, 5), (2, 4), (3, 7), (6, 8)] (ee=2, eo=0, oe=0, oo=2)
34. [(1, 5), (2, 7), (3, 4), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
35. [(1, 7), (2, 4), (3, 5), (6, 8)] (ee=2, eo=0, oe=0, oo=2)
36. [(1, 7), (2, 5), (3, 4), (6, 8)] (ee=1, eo=1, oe=1, oo=1)
37. [(1, 4), (2, 6), (3, 7), (5, 8)] (ee=1, eo=0, oe=2, oo=1)
38. [(1, 4), (2, 7), (3, 6), (5, 8)] (ee=0, eo=1, oe=3, oo=0)
39. [(1, 6), (2, 4), (3, 7), (5, 8)] (ee=1, eo=0, oe=2, oo=1)
40. [(1, 6), (2, 7), (3, 4), (5, 8)] (ee=0, eo=1, oe=3, oo=0)
41. [(1, 7), (2, 4), (3, 6), (5, 8)] (ee=1, eo=0, oe=2, oo=1)
42. [(1, 7), (2, 6), (3, 4), (5, 8)] (ee=1, eo=0, oe=2, oo=1)
43. [(1, 5), (2, 6), (3, 7), (4, 8)] (ee=2, eo=0, oe=0, oo=2)
44. [(1, 5), (2, 7), (3, 6), (4, 8)] (ee=1, eo=1, oe=1, oo=1)
45. [(1, 6), (2, 5), (3, 7), (4, 8)] (ee=1, eo=1, oe=1, oo=1)
46. [(1, 6), (2, 7), (3, 5), (4, 8)] (ee=1, eo=1, oe=1, oo=1)
47. [(1, 7), (2, 5), (3, 6), (4, 8)] (ee=1, eo=1, oe=1, oo=1)
48. [(1, 7), (2, 6), (3, 5), (4, 8)] (ee=2, eo=0, oe=0, oo=2)