Burhat Order

Burhat Order (back to Data page)

定義：

$w$ is one line permutation in $\mathcal{S}_n$

$l_O(w)= \Big| \{ ( w(i), w(j) ) \mid1\leq i < j \leq n, w(i)> w(j), i \not\equiv j ~ (mod ~2) \} \Big|$

$l_E(w)= \Big| \{ ( w(i), w(j) ) \mid1\leq i < j \leq n, w(i)> w(j), i \equiv j ~(mod ~2) \} \Big|$

$l_{ex}(w)= \Big| \{ ( w(i), w(j) ) \mid1\leq i < j \leq n, w(i)> w(j), i \equiv 0 ~(mod ~2) \} \Big|$

$l_{xe}(w)= \Big| \{ ( w(i), w(j) ) \mid1\leq i < j \leq n, w(i)> w(j), j \equiv 0 ~(mod ~2) \} \Big|$

$l_{ee}(w)= \Big| \{ ( w(i), w(j) ) \mid1\leq i < j \leq n, w(i)> w(j), i \equiv j \equiv 0 ~(mod ~2) \} \Big|$

目錄：

$l_O$
$l_E$
$l_{ex}$
$l_{xe}$
$l_{ee}$

$l_O$

n	total	$l_o$	numbers
3	6	0	1
		1	4
		2	1
4	24	0	1
		1	8
		2	6
		3	8
		4	1
5	120	0	1
		1	12
		2	23
		3	48
		4	23
		5	12
		6	1
6	720	0	1
		1	16
		2	59
		3	137
		4	147
		5	147
		6	137
		7	59
		8	16
		9	1
7	5040	0	1
		1	20
		2	113
		3	300
		4	631
		5	832
		6	1246
		7	832
		8	631
		9	300
		10	113
		11	20
		12	1
8	40320	0	1
		1	24
		2	183
		3	620
		4	1878
		5	2956
		6	5481
		7	5616
		8	6802
		9	5616
		10	5481
		11	2956
		12	1878
		13	620
		14	183
		15	24
		16	1

$l_E$

* $[reduced] = \frac{[number]}{1^{st}~ number~ of~ l_o ~in ~each ~n}$

n	total	$l_o$	numbers	reduced
3	6	0	3	1
3	6	1	3	1
4	24	0	6	1
		1	12	2
		2	6	1
5	120	0	10	1
		1	30	3
		2	40	4
		3	30	3
		4	10	1
6	720	0	20	1
		1	80	4
		2	160	8
		3	200	10
		4	160	8
		5	80	4
		6	20	1
7	5040	0	35	1
		1	175	5
		2	455	13
		3	805	23
		4	1050	30
		5	1050	30
		6	805	23
		7	455	13
		8	175	5
		9	35	1
8	40320	0	70	1
		1	420	6
		2	1330	19
		3	2940	42
		4	4970	71
		5	6720	96
		6	7420	106
		7	6720	96
		8	4970	71
		9	2940	42
		10	1330	19
		11	420	6
		12	70	1
9	362880	0	126	1
		1	882	7
		2	3276	26
		3	8568	68
		4	17514	139
		5	29484	234
		6	42084	334
		7	51786	411
		8	55440	440
		9	51786	411
		10	42084	334
		11	29484	234
		12	17514	139
		13	8568	68
		14	3276	26
		15	882	7
		16	126	1
10	3628800	0	252	1
		1	2016	8
		2	8568	34
		3	25704	102
		4	60732	241
		5	119448	474
		6	201852	801
		7	298872	1186
		8	392616	1558
		9	461160	1830
		10	486360	1930
		11	461160	1830
		12	392616	1558
		13	298872	1186
		14	201852	801
		15	119448	474
		16	60732	241
		17	25704	102
		18	8568	34
		19	2016	8
		20	252	1

$l_{ex}$

* $[reduced] = \frac{[number]}{(n-1)!!}$

n	total	$l_{ex}$	numbers	reduced
3	6	0	2	1
		1	2	1
		2	2	1
4	24	0	3	1
		1	6	2
		2	6	2
		3	6	2
		4	3	1
5	120	0	8	1
		1	16	2
		2	24	3
		3	24	3
		4	24	3
		5	16	2
		6	8	1
6	720	0	15	1
		1	45	3
		2	75	5
		3	105	7
		4	120	8
		5	120	8
		6	105	7
		7	75	5
		8	45	3
		9	15	1
7	5040	0	48	1
		1	144	3
		2	288	6
		3	432	9
		4	576	12
		5	672	14
		6	720	15
		7	672	14
		8	576	12
		9	432	9
		10	288	6
		11	144	3
		12	48	1
8	403420	0	105	1
		1	420	4
		2	945	9
		3	1680	16
		4	2520	24
		5	3360	32
		6	4095	39
		7	4620	44
		8	4830	46
		9	4620	44
		10	4095	39
		11	3360	32
		12	2520	24
		13	1680	16
		14	945	9
		15	420	4
		16	105	1
9	362880	0	384	1
		1	1536	4
		2	3840	10
		3	7296	19
		4	11904	31
		5	17280	45
		6	23040	60
		7	28416	74
		8	33024	86
		9	36096	94
		10	37248	97
		11	36096	94
		12	33024	86
		13	28416	74
		14	23040	60
		15	17280	45
		16	11904	31
		17	7296	19
		18	3840	10
		19	1536	4
		20	384	1
10	3628800	0	945	1
		1	4725	5
		2	13230	14
		3	28350	30
		4	51030	54
		5	81270	86
		6	118125	125
		7	159705	169
		8	203175	215
		9	244755	259
		10	280665	297
		11	307125	325
		12	321300	340
		13	321300	340
		14	307125	325
		15	280665	297
		16	244755	259
		17	203175	215
		18	159705	169
		19	118125	125
		20	81270	86
		21	51030	54
		22	28350	30
		23	13230	14
		24	4725	5
		25	945	1

$l_{xe}$

* $[reduced] = \frac{[number]}{(n-1)!!}$ if n is odd number
* $[reduced] = \frac{[number]}{(n)!!}$ if n is even number

n	total	$l_{xe}$	numbers	reduced
3	6	0	2	1
		1	2	1
		2	2	1
4	24	0	8	1
		1	8	1
		2	8	1
5	120	0	8	1
		1	16	2
		2	24	3
		3	24	3
		4	24	3
		5	16	2
		6	8	1
6	720	0	48	1
		1	96	2
		2	144	3
		3	144	3
		4	144	3
		5	96	2
		6	48	1
7	5040	0	48	1
		1	144	3
		2	288	6
		3	432	9
		4	576	12
		5	672	14
		6	720	15
		7	672	14
		8	576	12
		9	432	9
		10	288	6
		11	144	3
		12	48	1
8	40320	0	384	1
		1	1152	3
		2	2304	6
		3	3456	9
		4	4608	12
		5	5376	14
		6	5760	15
		7	5376	14
		8	4608	12
		9	3456	9
		10	2304	6
		11	1152	3
		12	384	1
9	362880	0	384	1
		1	1536	4
		2	3840	10
		3	7296	19
		4	11904	31
		5	17280	45
		6	23040	60
		7	28416	74
		8	33024	86
		9	36096	94
		10	37248	97
		11	36096	94
		12	33024	86
		13	28416	74
		14	23040	60
		15	17280	45
		16	11904	31
		17	7296	19
		18	3840	10
		19	1536	4
		20	384	1
10	3628800	0	3840	1
		1	15360	4
		2	38400	10
		3	72960	19
		4	119040	31
		5	172800	45
		6	230400	60
		7	284160	74
		8	330240	86
		9	360960	94
		10	372480	97
		11	360960	94
		12	330240	86
		13	284160	74
		14	230400	60
		15	172800	45
		16	119040	31
		17	72960	19
		18	38400	10
		19	15360	4
		20	3840	1

$l_{ee}$

n

total

$l_{ee}$

numbers