Perfect Matching of 2k

Perfect Matching of (2k), concordant, discordant, dissident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 (ds) = #$\{i \mid m_i > m_{i-1}+1 \}$

The statistic of [cc, dc, ds] seems equal to the statistic of [n-1-r, n-1-s, n-1-t], where it is the [asc-1, des-1, plat-1] for the Stirling Permutation
link

Google Colab

[r, s, t] = [n-1-cc, n-1-dc, n-1-ds]

n	total	[cc, dc, ds]	個數	[r, s, t]
2	3	[0, 0, 1]	1	[1, 1, 0]
		[0, 1, 0]	1	[1, 0, 1]
		[1, 0, 0]	1	[0, 1, 1]
3	15	[0, 0, 2]	1	[2, 2, 0]
		[0, 1, 1]	4	[2, 1, 1]
		[0, 2, 0]	1	[2, 0, 2]
		[1, 0, 1]	4	[1, 2, 1]
		[1, 1, 0]	4	[1, 1, 2]
		[2, 0, 0]	1	[0, 2, 2]
4	105	[0, 0, 3]	1	[3, 3, 0]
		[0, 1, 2]	11	[3, 2, 1]
		[0, 2, 1]	11	[3, 1, 2]
		[0, 3, 0]	1	[3, 0, 3]
		[1, 0, 2]	11	[2, 3, 1]
		[1, 1, 1]	36	[2, 2, 2]
		[1, 2, 0]	11	[2, 1, 3]
		[2, 0, 1]	11	[1, 3, 2]
		[2, 1, 0]	11	[1, 2, 3]
		[3, 0, 0]	1	[0, 3, 3]
5	945	[0, 0, 4]	1	[4, 4, 0]
		[0, 1, 3]	26	[4, 3, 1]
		[0, 2, 2]	66	[4, 2, 2]
		[0, 3, 1]	26	[4, 1, 3]
		[0, 4, 0]	1	[4, 0, 4]
		[1, 0, 3]	26	[3, 4, 1]
		[1, 1, 2]	196	[3, 3, 2]
		[1, 2, 1]	196	[3, 2, 3]
		[1, 3, 0]	26	[3, 1, 4]
		[2, 0, 2]	66	[2, 4, 2]
		[2, 1, 1]	196	[2, 3, 3]
		[2, 2, 0]	66	[2, 2, 4]
		[3, 0, 1]	26	[1, 4, 3]
		[3, 1, 0]	26	[1, 3, 4]
		[4, 0, 0]	1	[0, 4, 4]
6	10395	[0, 0, 5]	1	[5, 5, 0]
		[0, 1, 4]	57	[5, 4, 1]
		[0, 2, 3]	302	[5, 3, 2]
		[0, 3, 2]	302	[5, 2, 3]
		[0, 4, 1]	57	[5, 1, 4]
		[0, 5, 0]	1	[5, 0, 5]
		[1, 0, 4]	57	[4, 5, 1]
		[1, 1, 3]	848	[4, 4, 2]
		[1, 2, 2]	1898	[4, 3, 3]
		[1, 3, 1]	848	[4, 2, 4]
		[1, 4, 0]	57	[4, 1, 5]
		[2, 0, 3]	302	[3, 5, 2]
		[2, 1, 2]	1898	[3, 4, 3]
		[2, 2, 1]	1898	[3, 3, 4]
		[2, 3, 0]	302	[3, 2, 5]
		[3, 0, 2]	302	[2, 5, 3]
		[3, 1, 1]	848	[2, 4, 4]
		[3, 2, 0]	302	[2, 3, 5]
		[4, 0, 1]	57	[1, 5, 4]
		[4, 1, 0]	57	[1, 4, 5]
		[5, 0, 0]	1	[0, 5, 5]
7	135135	[0, 0, 6]	1	[6, 6, 0]
		[0, 1, 5]	120	[6, 5, 1]
		[0, 2, 4]	1191	[6, 4, 2]
		[0, 3, 3]	2416	[6, 3, 3]
		[0, 4, 2]	1191	[6, 2, 4]
		[0, 5, 1]	120	[6, 1, 5]
		[0, 6, 0]	1	[6, 0, 6]
		[1, 0, 5]	120	[5, 6, 1]
		[1, 1, 4]	3228	[5, 5, 2]
		[1, 2, 3]	13644	[5, 4, 3]
		[1, 3, 2]	13644	[5, 3, 4]
		[1, 4, 1]	3228	[5, 2, 5]
		[1, 5, 0]	120	[5, 1, 6]
		[2, 0, 4]	1191	[4, 6, 2]
		[2, 1, 3]	13644	[4, 5, 3]
		[2, 2, 2]	28470	[4, 4, 4]
		[2, 3, 1]	13644	[4, 3, 5]
		[2, 4, 0]	1191	[4, 2, 6]
		[3, 0, 3]	2416	[3, 6, 3]
		[3, 1, 2]	13644	[3, 5, 4]
		[3, 2, 1]	13644	[3, 4, 5]
		[3, 3, 0]	2416	[3, 3, 6]
		[4, 0, 2]	1191	[2, 6, 4]
		[4, 1, 1]	3228	[2, 5, 5]
		[4, 2, 0]	1191	[2, 4, 6]
		[5, 0, 1]	120	[1, 6, 5]
		[5, 1, 0]	120	[1, 5, 6]
		[6, 0, 0]	1	[0, 6, 6]