Perfect Matching of 2k

Perfect Matching of (2k), concordant, discordant, dissident_greater, dissident_smaller 的統計量 (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 $1=i_1 < i_2 < ... < i_n$.

$M=\{(m_1, M_1)(m_2, M_2)...(m_n, M_n)\mid m_1< m_2 <...< m_n\}$
- concordant (cc) = #$\{i \mid m_i = m_{i-1}+1 \mbox{ and } M_i > M_{i-1}\}$
- discordant (dc) = #$\{i \mid m_i = m_{i-1}+1 \mbox{ and } M_i < M_{i-1}\}$
- dissident_greater (dsg) = #$\{i \mid m_i > m_{i-1}+1 \mbox{ and } M_i > M_{i-1}\}$
- dissident_smaller (dss) = #$\{i \mid m_i > m_{i-1}+1 \mbox{ and } M_i < M_{i-1}\}$

The statistic of [cc, dc, dsg, dss] seems equal to the statistic of [n-1-r, n-1-s, n-1-t, n-1-p], if we combine dog and dss, we can get the following
link

Google Colab

[r, s, t, p] = [n-1-cc, n-1-dc, n-1-dsg, n-1-dss]

n	total	[cc, dc, dsg, dss]	個數	[r, s, t, p]
2	3	[0, 0, 1, 0]	1	[1, 1, 0, 1]
		[0, 1, 0, 0]	1	[1, 0, 1, 1]
		[1, 0, 0, 0]	1	[0, 1, 1, 1]
3	15	[0, 0, 2, 0]	1	[2, 2, 0, 2]
		[0, 1, 1, 0]	4	[2, 1, 1, 2]
		[0, 2, 0, 0]	1	[2, 0, 2, 2]
		[1, 0, 0, 1]	1	[1, 2, 2, 1]
		[1, 0, 1, 0]	3	[1, 2, 1, 2]
		[1, 1, 0, 0]	4	[1, 1, 2, 2]
		[2, 0, 0, 0]	1	[0, 2, 2, 2]
4	105	[0, 0, 3, 0]	1	[3, 3, 0, 3]
		[0, 1, 1, 1]	1	[3, 2, 2, 2]
		[0, 1, 2, 0]	10	[3, 2, 1, 3]
		[0, 2, 1, 0]	11	[3, 1, 2, 3]
		[0, 3, 0, 0]	1	[3, 0, 3, 3]
		[1, 0, 1, 1]	5	[2, 3, 2, 2]
		[1, 0, 2, 0]	6	[2, 3, 1, 3]
		[1, 1, 0, 1]	6	[2, 2, 3, 2]
		[1, 1, 1, 0]	30	[2, 2, 2, 3]
		[1, 2, 0, 0]	11	[2, 1, 3, 3]
		[2, 0, 0, 1]	5	[1, 3, 3, 2]
		[2, 0, 1, 0]	6	[1, 3, 2, 3]
		[2, 1, 0, 0]	11	[1, 2, 3, 3]
		[3, 0, 0, 0]	1	[0, 3, 3, 3]
5	945	[0, 0, 4, 0]	1	[4, 4, 0, 4]
		[0, 1, 2, 1]	6	[4, 3, 2, 3]
		[0, 1, 3, 0]	20	[4, 3, 1, 4]
		[0, 2, 1, 1]	7	[4, 2, 3, 3]
		[0, 2, 2, 0]	59	[4, 2, 2, 4]
		[0, 3, 1, 0]	26	[4, 1, 3, 4]
		[0, 4, 0, 0]	1	[4, 0, 4, 4]
		[1, 0, 1, 2]	1	[3, 4, 3, 2]
		[1, 0, 2, 1]	15	[3, 4, 2, 3]
		[1, 0, 3, 0]	10	[3, 4, 1, 4]
		[1, 1, 0, 2]	1	[3, 3, 4, 2]
		[1, 1, 1, 1]	73	[3, 3, 3, 3]
		[1, 1, 2, 0]	122	[3, 3, 2, 4]
		[1, 2, 0, 1]	23	[3, 2, 4, 3]
		[1, 2, 1, 0]	173	[3, 2, 3, 4]
		[1, 3, 0, 0]	26	[3, 1, 4, 4]
		[2, 0, 0, 2]	5	[2, 4, 4, 2]
		[2, 0, 1, 1]	41	[2, 4, 3, 3]
		[2, 0, 2, 0]	20	[2, 4, 2, 4]
		[2, 1, 0, 1]	63	[2, 3, 4, 3]
		[2, 1, 1, 0]	133	[2, 3, 3, 4]
		[2, 2, 0, 0]	66	[2, 2, 4, 4]
		[3, 0, 0, 1]	16	[1, 4, 4, 3]
		[3, 0, 1, 0]	10	[1, 4, 3, 4]
		[3, 1, 0, 0]	26	[1, 3, 4, 4]
		[4, 0, 0, 0]	1	[0, 4, 4, 4]
6	10395	[0, 0, 5, 0]	1	[5, 5, 0, 5]
		[0, 1, 2, 2]	1	[5, 4, 3, 3]
		[0, 1, 3, 1]	21	[5, 4, 2, 4]
		[0, 1, 4, 0]	35	[5, 4, 1, 5]
		[0, 2, 1, 2]	1	[5, 3, 4, 3]
		[0, 2, 2, 1]	82	[5, 3, 3, 4]
		[0, 2, 3, 0]	219	[5, 3, 2, 5]
		[0, 3, 1, 1]	30	[5, 2, 4, 4]
		[0, 3, 2, 0]	272	[5, 2, 3, 5]
		[0, 4, 1, 0]	57	[5, 1, 4, 5]
		[0, 5, 0, 0]	1	[5, 0, 5, 5]
		[1, 0, 2, 2]	7	[4, 5, 3, 3]
		[1, 0, 3, 1]	35	[4, 5, 2, 4]
		[1, 0, 4, 0]	15	[4, 5, 1, 5]
		[1, 1, 1, 2]	45	[4, 4, 4, 3]
		[1, 1, 2, 1]	439	[4, 4, 3, 4]
		[1, 1, 3, 0]	364	[4, 4, 2, 5]
		[1, 2, 0, 2]	9	[4, 3, 5, 3]
		[1, 2, 1, 1]	592	[4, 3, 4, 4]
		[1, 2, 2, 0]	1297	[4, 3, 3, 5]
		[1, 3, 0, 1]	72	[4, 2, 5, 4]
		[1, 3, 1, 0]	776	[4, 2, 4, 5]
		[1, 4, 0, 0]	57	[4, 1, 5, 5]
		[2, 0, 0, 3]	1	[3, 5, 5, 2]
		[2, 0, 1, 2]	67	[3, 5, 4, 3]
		[2, 0, 2, 1]	184	[3, 5, 3, 4]
		[2, 0, 3, 0]	50	[3, 5, 2, 5]
		[2, 1, 0, 2]	92	[3, 4, 5, 3]
		[2, 1, 1, 1]	1006	[3, 4, 4, 4]
		[2, 1, 2, 0]	800	[3, 4, 3, 5]
		[2, 2, 0, 1]	454	[3, 3, 5, 4]
		[2, 2, 1, 0]	1444	[3, 3, 4, 5]
		[2, 3, 0, 0]	302	[3, 2, 5, 5]
		[3, 0, 0, 2]	56	[2, 5, 5, 3]
		[3, 0, 1, 1]	196	[2, 5, 4, 4]
		[3, 0, 2, 0]	50	[2, 5, 3, 5]
		[3, 1, 0, 1]	386	[2, 4, 5, 4]
		[3, 1, 1, 0]	462	[2, 4, 4, 5]
		[3, 2, 0, 0]	302	[2, 3, 5, 5]
		[4, 0, 0, 1]	42	[1, 5, 5, 4]
		[4, 0, 1, 0]	15	[1, 5, 4, 5]
		[4, 1, 0, 0]	57	[1, 4, 5, 5]
		[5, 0, 0, 0]	1	[0, 5, 5, 5]
7	135135	[0, 0, 6, 0]	1	[6, 6, 0, 6]
		[0, 1, 3, 2]	8	[6, 5, 3, 4]
		[0, 1, 4, 1]	56	[6, 5, 2, 5]
		[0, 1, 5, 0]	56	[6, 5, 1, 6]
		[0, 2, 2, 2]	42	[6, 4, 4, 4]
		[0, 2, 3, 1]	503	[6, 4, 3, 5]
		[0, 2, 4, 0]	646	[6, 4, 2, 6]
		[0, 3, 1, 2]	10	[6, 3, 5, 4]
		[0, 3, 2, 1]	644	[6, 3, 4, 5]
		[0, 3, 3, 0]	1762	[6, 3, 3, 6]
		[0, 4, 1, 1]	102	[6, 2, 5, 5]
		[0, 4, 2, 0]	1089	[6, 2, 4, 6]
		[0, 5, 1, 0]	120	[6, 1, 5, 6]
		[0, 6, 0, 0]	1	[6, 0, 6, 6]
		[1, 0, 2, 3]	1	[5, 6, 4, 3]
		[1, 0, 3, 2]	28	[5, 6, 3, 4]
		[1, 0, 4, 1]	70	[5, 6, 2, 5]
		[1, 0, 5, 0]	21	[5, 6, 1, 6]
		[1, 1, 1, 3]	11	[5, 5, 5, 3]
		[1, 1, 2, 2]	498	[5, 5, 4, 4]
		[1, 1, 3, 1]	1823	[5, 5, 3, 5]
		[1, 1, 4, 0]	896	[5, 5, 2, 6]
		[1, 2, 0, 3]	1	[5, 4, 6, 3]
		[1, 2, 1, 2]	647	[5, 4, 5, 4]
		[1, 2, 2, 1]	6468	[5, 4, 4, 5]
		[1, 2, 3, 0]	6528	[5, 4, 3, 6]
		[1, 3, 0, 2]	48	[5, 3, 6, 4]
		[1, 3, 1, 1]	3578	[5, 3, 5, 5]
		[1, 3, 2, 0]	10018	[5, 3, 4, 6]
		[1, 4, 0, 1]	201	[5, 2, 6, 5]
		[1, 4, 1, 0]	3027	[5, 2, 5, 6]
		[1, 5, 0, 0]	120	[5, 1, 6, 6]
		[2, 0, 1, 3]	39	[4, 6, 5, 3]
		[2, 0, 2, 2]	443	[4, 6, 4, 4]
		[2, 0, 3, 1]	604	[4, 6, 3, 5]
		[2, 0, 4, 0]	105	[4, 6, 2, 6]
		[2, 1, 0, 3]	48	[4, 5, 6, 3]
		[2, 1, 1, 2]	2335	[4, 5, 5, 4]
		[2, 1, 2, 1]	7930	[4, 5, 4, 5]
		[2, 1, 3, 0]	3331	[4, 5, 3, 6]
		[2, 2, 0, 2]	913	[4, 4, 6, 4]
		[2, 2, 1, 1]	12990	[4, 4, 5, 5]
		[2, 2, 2, 0]	14567	[4, 4, 4, 6]
		[2, 3, 0, 1]	2498	[4, 3, 6, 5]
		[2, 3, 1, 0]	11146	[4, 3, 5, 6]
		[2, 4, 0, 0]	1191	[4, 2, 6, 6]
		[3, 0, 0, 3]	71	[3, 6, 6, 3]
		[3, 0, 1, 2]	936	[3, 6, 5, 4]
		[3, 0, 2, 1]	1234	[3, 6, 4, 5]
		[3, 0, 3, 0]	175	[3, 6, 3, 6]
		[3, 1, 0, 2]	1598	[3, 5, 6, 4]
		[3, 1, 1, 1]	8190	[3, 5, 5, 5]
		[3, 1, 2, 0]	3856	[3, 5, 4, 6]
		[3, 2, 0, 1]	4804	[3, 4, 6, 5]
		[3, 2, 1, 0]	8840	[3, 4, 5, 6]
		[3, 3, 0, 0]	2416	[3, 3, 6, 6]
		[4, 0, 0, 2]	364	[2, 6, 6, 4]
		[4, 0, 1, 1]	722	[2, 6, 5, 5]
		[4, 0, 2, 0]	105	[2, 6, 4, 6]
		[4, 1, 0, 1]	1830	[2, 5, 6, 5]
		[4, 1, 1, 0]	1398	[2, 5, 5, 6]
		[4, 2, 0, 0]	1191	[2, 4, 6, 6]
		[5, 0, 0, 1]	99	[1, 6, 6, 5]
		[5, 0, 1, 0]	21	[1, 6, 5, 6]
[5, 1, 0, 0]	120	[1, 5, 6, 6]
[6, 0, 0, 0]	1	[0, 6, 6, 6]
8	2027025	[0, 0, 7, 0]	1	[7, 7, 0, 7]
		[0, 1, 3, 3]	1	[7, 6, 4, 4]
		[0, 1, 4, 2]	36	[7, 6, 3, 5]
		[0, 1, 5, 1]	126	[7, 6, 2, 6]
		[0, 1, 6, 0]	84	[7, 6, 1, 7]
		[0, 2, 2, 3]	10	[7, 5, 5, 4]
		[0, 2, 3, 2]	477	[7, 5, 4, 5]
		[0, 2, 4, 1]	2180	[7, 5, 3, 6]
		[0, 2, 5, 0]	1626	[7, 5, 2, 7]
		[0, 3, 1, 3]	1	[7, 4, 6, 4]
		[0, 3, 2, 2]	603	[7, 4, 5, 5]
		[0, 3, 3, 1]	6604	[7, 4, 4, 6]
		[0, 3, 4, 0]	8411	[7, 4, 3, 7]
		[0, 4, 1, 2]	58	[7, 3, 6, 5]
		[0, 4, 2, 1]	3840	[7, 3, 5, 6]
		[0, 4, 3, 0]	11721	[7, 3, 4, 7]
		[0, 5, 1, 1]	303	[7, 2, 6, 6]
		[0, 5, 2, 0]	3990	[7, 2, 5, 7]
		[0, 6, 1, 0]	247	[7, 1, 6, 7]
		[0, 7, 0, 0]	1	[7, 0, 7, 7]
		[1, 0, 3, 3]	9	[6, 7, 4, 4]
		[1, 0, 4, 2]	84	[6, 7, 3, 5]
		[1, 0, 5, 1]	126	[6, 7, 2, 6]
		[1, 0, 6, 0]	28	[6, 7, 1, 7]
		[1, 1, 1, 4]	1	[6, 6, 6, 3]
		[1, 1, 2, 3]	275	[6, 6, 5, 4]
		[1, 1, 3, 2]	3175	[6, 6, 4, 5]
		[1, 1, 4, 1]	5981	[6, 6, 3, 6]
		[1, 1, 5, 0]	1932	[6, 6, 2, 7]
		[1, 2, 1, 3]	339	[6, 5, 6, 4]
		[1, 2, 2, 2]	11940	[6, 5, 5, 5]
		[1, 2, 3, 1]	44598	[6, 5, 4, 6]
		[1, 2, 4, 0]	25404	[6, 5, 3, 7]
		[1, 3, 0, 3]	12	[6, 4, 7, 4]
		[1, 3, 1, 2]	5857	[6, 4, 6, 5]
		[1, 3, 2, 1]	66439	[6, 4, 5, 6]
		[1, 3, 3, 0]	81044	[6, 4, 4, 7]
		[1, 4, 0, 2]	198	[6, 3, 7, 5]
		[1, 4, 1, 1]	18111	[6, 3, 6, 6]
		[1, 4, 2, 0]	63972	[6, 3, 5, 7]
		[1, 5, 0, 1]	522	[6, 2, 7, 6]
		[1, 5, 1, 0]	10842	[6, 2, 6, 7]
		[1, 6, 0, 0]	247	[6, 1, 7, 7]
		[2, 0, 1, 4]	10	[5, 7, 6, 3]
		[2, 0, 2, 3]	449	[5, 7, 5, 4]
		[2, 0, 3, 2]	2012	[5, 7, 4, 5]
		[2, 0, 4, 1]	1626	[5, 7, 3, 6]
		[2, 0, 5, 0]	196	[5, 7, 2, 7]
		[2, 1, 0, 4]	11	[5, 6, 7, 3]
		[2, 1, 1, 3]	2497	[5, 6, 6, 4]
		[2, 1, 2, 2]	26301	[5, 6, 5, 5]
		[2, 1, 3, 1]	42502	[5, 6, 4, 6]
		[2, 1, 4, 0]	10970	[5, 6, 3, 7]
		[2, 2, 0, 3]	797	[5, 5, 7, 4]
		[2, 2, 1, 2]	40451	[5, 5, 6, 5]
		[2, 2, 2, 1]	168924	[5, 5, 5, 6]
		[2, 2, 3, 0]	95906	[5, 5, 4, 7]
		[2, 3, 0, 2]	6605	[5, 4, 7, 5]
		[2, 3, 1, 1]	120580	[5, 4, 6, 6]
		[2, 3, 2, 0]	178893	[5, 4, 5, 7]
		[2, 4, 0, 1]	11727	[5, 3, 7, 6]
		[2, 4, 1, 0]	70554	[5, 3, 6, 7]
		[2, 5, 0, 0]	4293	[5, 2, 7, 7]
		[3, 0, 0, 4]	39	[4, 7, 7, 3]
[3, 0, 1, 3]	1852	[4, 7, 6, 4]
[3, 0, 2, 2]	7803	[4, 7, 5, 5]
[3, 0, 3, 1]	5435	[4, 7, 4, 6]
[3, 0, 4, 0]	490	[4, 7, 3, 7]
[3, 1, 0, 3]	2859	[4, 6, 7, 4]
[3, 1, 1, 2]	45383	[4, 6, 6, 5]
[3, 1, 2, 1]	83740	[4, 6, 5, 6]
[3, 1, 3, 0]	21370	[4, 6, 4, 7]
[3, 2, 0, 2]	23947	[4, 5, 7, 5]
[3, 2, 1, 1]	166204	[4, 5, 6, 6]
[3, 2, 2, 0]	115927	[4, 5, 5, 7]
[3, 3, 0, 1]	42852	[4, 4, 7, 6]
[3, 3, 1, 0]	110500	[4, 4, 6, 7]
[3, 4, 0, 0]	15619	[4, 3, 7, 7]
[4, 0, 0, 3]	1220	[3, 7, 7, 4]
[4, 0, 1, 2]	7818	[3, 7, 6, 5]
[4, 0, 2, 1]	6091	[3, 7, 5, 6]
[4, 0, 3, 0]	490	[3, 7, 4, 7]
[4, 1, 0, 2]	16402	[3, 6, 7, 5]
[4, 1, 1, 1]	50419	[3, 6, 6, 6]
[4, 1, 2, 0]	15460	[3, 6, 5, 7]
[4, 2, 0, 1]	37206	[3, 5, 7, 6]
[4, 2, 1, 0]	45075	[3, 5, 6, 7]
[4, 3, 0, 0]	15619	[3, 4, 7, 7]
[5, 0, 0, 2]	1814	[2, 7, 7, 5]
[5, 0, 1, 1]	2283	[2, 7, 6, 6]
[5, 0, 2, 0]	196	[2, 7, 5, 7]
[5, 1, 0, 1]	7482	[2, 6, 7, 6]
[5, 1, 1, 0]	3882	[2, 6, 6, 7]
[5, 2, 0, 0]	4293	[2, 5, 7, 7]
[6, 0, 0, 1]	219	[1, 7, 7, 6]
[6, 0, 1, 0]	28	[1, 7, 6, 7]
[6, 1, 0, 0]	247	[1, 6, 7, 7]
[7, 0, 0, 0]	1	[0, 7, 7, 7]