Identifier
-
Mp00323:
Integer partitions
—Loehr-Warrington inverse⟶
Integer partitions
St001936: Integer partitions ⟶ ℤ
Values
[1] => [1] => 1
[2] => [1,1] => 1
[1,1] => [2] => 1
[3] => [2,1] => 2
[2,1] => [1,1,1] => 4
[1,1,1] => [3] => 1
[4] => [2,2] => 4
[3,1] => [1,1,1,1] => 30
[2,2] => [2,1,1] => 10
[2,1,1] => [3,1] => 3
[1,1,1,1] => [4] => 1
[5] => [3,2] => 6
[4,1] => [3,1,1] => 18
[3,2] => [1,1,1,1,1] => 336
[3,1,1] => [2,1,1,1] => 84
[2,2,1] => [2,2,1] => 24
[2,1,1,1] => [4,1] => 4
[1,1,1,1,1] => [5] => 1
[6] => [3,3] => 9
[5,1] => [3,2,1] => 42
[4,2] => [2,1,1,1,1] => 1008
[4,1,1] => [2,2,1,1] => 224
[3,3] => [3,1,1,1] => 168
[3,2,1] => [1,1,1,1,1,1] => 5040
[3,1,1,1] => [4,1,1] => 28
[2,2,2] => [2,2,2] => 56
[2,2,1,1] => [4,2] => 8
[2,1,1,1,1] => [5,1] => 5
[1,1,1,1,1,1] => [6] => 1
[7] => [4,3] => 12
[6,1] => [3,3,1] => 72
[5,2] => [3,2,1,1] => 432
[5,1,1] => [4,2,1] => 64
[4,3] => [2,2,1,1,1] => 2880
[4,2,1] => [1,1,1,1,1,1,1] => 95040
[4,1,1,1] => [2,2,2,1] => 576
[3,3,1] => [2,1,1,1,1,1] => 15840
[3,2,2] => [3,1,1,1,1] => 2160
[3,2,1,1] => [4,1,1,1] => 288
[3,1,1,1,1] => [5,1,1] => 40
[2,2,2,1] => [3,2,2] => 96
[2,2,1,1,1] => [5,2] => 10
[2,1,1,1,1,1] => [6,1] => 6
[1,1,1,1,1,1,1] => [7] => 1
[8] => [4,4] => 16
[7,1] => [4,3,1] => 108
[6,2] => [3,3,1,1] => 810
[6,1,1] => [4,2,2] => 144
[5,3] => [2,2,2,1,1] => 7920
[5,2,1] => [4,1,1,1,1] => 3960
[5,1,1,1] => [2,2,2,2] => 1440
[4,4] => [4,2,1,1] => 720
[4,3,1] => [1,1,1,1,1,1,1,1] => 2162160
[4,2,2] => [2,1,1,1,1,1,1] => 308880
[4,2,1,1] => [3,1,1,1,1,1] => 35640
[4,1,1,1,1] => [5,2,1] => 90
[3,3,2] => [2,2,1,1,1,1] => 47520
[3,3,1,1] => [3,2,1,1,1] => 5940
[3,2,2,1] => [3,2,2,1] => 1080
[3,2,1,1,1] => [5,1,1,1] => 450
[3,1,1,1,1,1] => [6,1,1] => 54
[2,2,2,2] => [3,3,2] => 162
[2,2,2,1,1] => [5,3] => 15
[2,2,1,1,1,1] => [6,2] => 12
[2,1,1,1,1,1,1] => [7,1] => 7
[1,1,1,1,1,1,1,1] => [8] => 1
[9] => [5,4] => 20
[8,1] => [4,4,1] => 160
[7,2] => [4,3,1,1] => 1320
[7,1,1] => [4,3,2] => 240
[6,3] => [4,2,2,1] => 1760
[6,2,1] => [4,2,1,1,1] => 10560
[6,1,1,1] => [5,2,2] => 200
[5,4] => [2,2,2,2,1] => 21120
[5,3,1] => [3,1,1,1,1,1,1] => 720720
[5,2,2] => [3,2,1,1,1,1] => 102960
[5,2,1,1] => [3,3,1,1,1] => 11880
[5,1,1,1,1] => [3,2,2,2] => 2640
[4,4,1] => [4,1,1,1,1,1] => 68640
[4,3,2] => [1,1,1,1,1,1,1,1,1] => 57657600
[4,3,1,1] => [2,1,1,1,1,1,1,1] => 7207200
[4,2,2,1] => [2,2,1,1,1,1,1] => 960960
[4,2,1,1,1] => [5,1,1,1,1] => 6600
[4,1,1,1,1,1] => [6,2,1] => 120
[3,3,3] => [3,2,2,1,1] => 15840
[3,3,2,1] => [2,2,2,1,1,1] => 137280
[3,3,1,1,1] => [5,2,1,1] => 1100
[3,2,2,2] => [3,3,2,1] => 1980
[3,2,2,1,1] => [5,3,1] => 150
[3,2,1,1,1,1] => [6,1,1,1] => 660
[3,1,1,1,1,1,1] => [7,1,1] => 70
[2,2,2,2,1] => [3,3,3] => 270
[2,2,2,1,1,1] => [6,3] => 18
[2,2,1,1,1,1,1] => [7,2] => 14
[2,1,1,1,1,1,1,1] => [8,1] => 8
[1,1,1,1,1,1,1,1,1] => [9] => 1
[10] => [5,5] => 25
[9,1] => [5,4,1] => 220
[8,2] => [4,4,1,1] => 2112
[8,1,1] => [4,4,2] => 352
[7,3] => [4,3,2,1] => 3168
>>> Load all 989 entries. <<<[7,2,1] => [4,3,1,1,1] => 20592
[7,1,1,1] => [5,3,2] => 330
[6,4] => [2,2,2,2,2] => 54912
[6,3,1] => [4,2,1,1,1,1] => 192192
[6,2,2] => [3,3,2,1,1] => 30888
[6,2,1,1] => [5,3,1,1] => 1980
[6,1,1,1,1] => [3,3,2,2] => 4752
[5,5] => [4,2,2,2] => 4224
[5,4,1] => [3,3,1,1,1,1] => 216216
[5,3,2] => [2,1,1,1,1,1,1,1,1] => 196035840
[5,3,1,1] => [2,2,1,1,1,1,1,1] => 23063040
[5,2,2,1] => [2,2,2,1,1,1,1] => 2882880
[5,2,1,1,1] => [3,2,2,2,1] => 41184
[5,1,1,1,1,1] => [6,2,2] => 264
[4,4,2] => [3,1,1,1,1,1,1,1] => 17297280
[4,4,1,1] => [3,2,1,1,1,1,1] => 2162160
[4,3,3] => [4,1,1,1,1,1,1] => 1441440
[4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => 1764322560
[4,3,1,1,1] => [5,1,1,1,1,1] => 120120
[4,2,2,2] => [3,2,2,1,1,1] => 288288
[4,2,2,1,1] => [5,2,1,1,1] => 17160
[4,2,1,1,1,1] => [6,1,1,1,1] => 10296
[4,1,1,1,1,1,1] => [7,2,1] => 154
[3,3,3,1] => [2,2,2,2,1,1] => 384384
[3,3,2,2] => [4,2,2,1,1] => 27456
[3,3,2,1,1] => [5,2,2,1] => 2640
[3,3,1,1,1,1] => [6,2,1,1] => 1584
[3,2,2,2,1] => [3,3,3,1] => 3564
[3,2,2,1,1,1] => [6,3,1] => 198
[3,2,1,1,1,1,1] => [7,1,1,1] => 924
[3,1,1,1,1,1,1,1] => [8,1,1] => 88
[2,2,2,2,2] => [4,3,3] => 396
[2,2,2,2,1,1] => [6,4] => 24
[2,2,2,1,1,1,1] => [7,3] => 21
[2,2,1,1,1,1,1,1] => [8,2] => 16
[2,1,1,1,1,1,1,1,1] => [9,1] => 9
[1,1,1,1,1,1,1,1,1,1] => [10] => 1
[11] => [6,5] => 30
[10,1] => [5,5,1] => 300
[9,2] => [5,4,1,1] => 3120
[9,1,1] => [5,4,2] => 480
[8,3] => [4,4,2,1] => 4992
[8,2,1] => [4,4,1,1,1] => 34944
[8,1,1,1] => [5,3,3] => 540
[7,4] => [5,2,2,2] => 6240
[7,3,1] => [4,3,1,1,1,1] => 393120
[7,2,2] => [4,3,2,1,1] => 52416
[7,2,1,1] => [5,3,2,1] => 4680
[7,1,1,1,1] => [6,3,2] => 432
[6,5] => [3,2,2,2,2] => 104832
[6,4,1] => [3,2,2,2,1,1] => 786240
[6,3,2] => [4,2,1,1,1,1,1] => 4193280
[6,3,1,1] => [5,2,1,1,1,1] => 327600
[6,2,2,1] => [5,2,2,1,1] => 43680
[6,2,1,1,1] => [3,3,2,2,1] => 78624
[6,1,1,1,1,1] => [3,3,3,2] => 8424
[5,5,1] => [5,3,1,1,1] => 32760
[5,4,2] => [2,2,1,1,1,1,1,1,1] => 641571840
[5,4,1,1] => [2,2,2,1,1,1,1,1] => 71285760
[5,3,3] => [3,2,1,1,1,1,1,1] => 53464320
[5,3,1,1,1] => [3,2,2,1,1,1,1] => 6289920
[5,2,2,2] => [2,2,2,2,1,1,1] => 8386560
[5,2,2,1,1] => [4,2,2,1,1,1] => 524160
[5,2,1,1,1,1] => [6,3,1,1] => 2808
[5,1,1,1,1,1,1] => [7,2,2] => 336
[4,4,3] => [3,3,1,1,1,1,1] => 4717440
[4,4,1,1,1] => [3,3,2,1,1,1] => 589680
[4,3,3,1] => [3,1,1,1,1,1,1,1,1] => 481178880
[4,3,2,2] => [4,1,1,1,1,1,1,1] => 35642880
[4,3,2,1,1] => [5,1,1,1,1,1,1] => 2620800
[4,3,1,1,1,1] => [6,1,1,1,1,1] => 196560
[4,2,2,2,1] => [3,3,3,1,1] => 58968
[4,2,2,1,1,1] => [6,2,1,1,1] => 26208
[4,2,1,1,1,1,1] => [7,1,1,1,1] => 15288
[4,1,1,1,1,1,1,1] => [8,2,1] => 192
[3,3,3,2] => [2,2,2,2,2,1] => 1048320
[3,3,3,1,1] => [4,2,2,2,1] => 69888
[3,3,2,2,1] => [4,3,2,2] => 7488
[3,3,2,1,1,1] => [6,2,2,1] => 3744
[3,3,1,1,1,1,1] => [7,2,1,1] => 2184
[3,2,2,2,2] => [4,3,3,1] => 5616
[3,2,2,2,1,1] => [6,4,1] => 288
[3,2,2,1,1,1,1] => [7,3,1] => 252
[3,2,1,1,1,1,1,1] => [8,1,1,1] => 1248
[3,1,1,1,1,1,1,1,1] => [9,1,1] => 108
[2,2,2,2,2,1] => [4,4,3] => 576
[2,2,2,2,1,1,1] => [7,4] => 28
[2,2,2,1,1,1,1,1] => [8,3] => 24
[2,2,1,1,1,1,1,1,1] => [9,2] => 18
[2,1,1,1,1,1,1,1,1,1] => [10,1] => 10
[1,1,1,1,1,1,1,1,1,1,1] => [11] => 1
[12] => [6,6] => 36
[11,1] => [6,5,1] => 390
[10,2] => [5,5,1,1] => 4550
[10,1,1] => [5,5,2] => 650
[9,3] => [5,4,2,1] => 7280
[9,2,1] => [5,4,1,1,1] => 54600
[9,1,1,1] => [5,4,3] => 780
[8,4] => [4,4,2,2] => 11648
[8,3,1] => [4,4,1,1,1,1] => 698880
[8,2,2] => [4,4,2,1,1] => 87360
[8,2,1,1] => [5,3,3,1] => 8190
[8,1,1,1,1] => [6,3,3] => 702
[7,5] => [3,3,2,2,2] => 196560
[7,4,1] => [5,3,2,1,1] => 81900
[7,3,2] => [4,3,1,1,1,1,1] => 8910720
[7,3,1,1] => [3,3,2,2,1,1] => 1572480
[7,2,2,1] => [5,2,2,2,1] => 109200
[7,2,1,1,1] => [6,3,2,1] => 6552
[7,1,1,1,1,1] => [3,3,3,3] => 14742
[6,6] => [5,3,2,2] => 10920
[6,5,1] => [3,2,2,2,2,1] => 2096640
[6,4,2] => [3,2,2,1,1,1,1,1] => 160392960
[6,4,1,1] => [3,2,2,2,1,1,1] => 17821440
[6,3,3] => [3,3,2,1,1,1,1] => 13366080
[6,3,2,1] => [5,1,1,1,1,1,1,1] => 66830400
[6,3,1,1,1] => [4,3,2,1,1,1] => 1048320
[6,2,2,2] => [4,2,2,2,1,1] => 1397760
[6,2,2,1,1] => [4,3,2,2,1] => 131040
[6,2,1,1,1,1] => [3,3,3,2,1] => 147420
[6,1,1,1,1,1,1] => [7,3,2] => 546
[5,5,2] => [5,2,1,1,1,1,1] => 7425600
[5,5,1,1] => [5,2,2,1,1,1] => 873600
[5,4,3] => [2,2,2,1,1,1,1,1,1] => 2031644160
[5,4,1,1,1] => [2,2,2,2,1,1,1,1] => 213857280
[5,3,2,1,1] => [4,1,1,1,1,1,1,1,1] => 1015822080
[5,3,1,1,1,1] => [6,2,1,1,1,1] => 524160
[5,2,2,2,1] => [2,2,2,2,2,1,1] => 23761920
[5,2,2,1,1,1] => [6,2,2,1,1] => 65520
[5,2,1,1,1,1,1] => [7,3,1,1] => 3822
[5,1,1,1,1,1,1,1] => [8,2,2] => 416
[4,4,4] => [5,3,1,1,1,1] => 655200
[4,4,2,2] => [3,2,1,1,1,1,1,1,1] => 1523733120
[4,4,2,1,1] => [4,2,1,1,1,1,1,1] => 106928640
[4,4,1,1,1,1] => [6,3,1,1,1] => 49140
[4,3,3,2] => [3,3,1,1,1,1,1,1] => 120294720
[4,3,3,1,1] => [4,2,2,1,1,1,1] => 11880960
[4,3,2,2,1] => [3,3,3,1,1,1] => 1179360
[4,3,2,1,1,1] => [6,1,1,1,1,1,1] => 4455360
[4,3,1,1,1,1,1] => [7,1,1,1,1,1] => 305760
[4,2,2,2,2] => [4,3,3,1,1] => 98280
[4,2,2,2,1,1] => [6,4,1,1] => 4368
[4,2,2,1,1,1,1] => [7,2,1,1,1] => 38220
[4,2,1,1,1,1,1,1] => [8,1,1,1,1] => 21840
[4,1,1,1,1,1,1,1,1] => [9,2,1] => 234
[3,3,3,3] => [4,2,2,2,2] => 174720
[3,3,3,2,1] => [2,2,2,2,2,2] => 2795520
[3,3,3,1,1,1] => [6,2,2,2] => 8736
[3,3,2,2,2] => [4,3,3,2] => 13104
[3,3,2,2,1,1] => [6,4,2] => 624
[3,3,2,1,1,1,1] => [7,2,2,1] => 5096
[3,3,1,1,1,1,1,1] => [8,2,1,1] => 2912
[3,2,2,2,2,1] => [4,4,3,1] => 8736
[3,2,2,2,1,1,1] => [7,4,1] => 364
[3,2,2,1,1,1,1,1] => [8,3,1] => 312
[3,2,1,1,1,1,1,1,1] => [9,1,1,1] => 1638
[3,1,1,1,1,1,1,1,1,1] => [10,1,1] => 130
[2,2,2,2,2,2] => [4,4,4] => 832
[2,2,2,2,2,1,1] => [7,5] => 35
[2,2,2,2,1,1,1,1] => [8,4] => 32
[2,2,2,1,1,1,1,1,1] => [9,3] => 27
[2,2,1,1,1,1,1,1,1,1] => [10,2] => 20
[2,1,1,1,1,1,1,1,1,1,1] => [11,1] => 11
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => 1
[13] => [7,6] => 42
[12,1] => [6,6,1] => 504
[11,2] => [6,5,1,1] => 6300
[11,1,1] => [6,5,2] => 840
[10,3] => [5,5,2,1] => 10500
[10,2,1] => [5,5,1,1,1] => 84000
[10,1,1,1] => [5,5,3] => 1050
[9,4] => [5,4,2,2] => 16800
[9,3,1] => [5,4,1,1,1,1] => 1142400
[9,2,2] => [5,4,2,1,1] => 134400
[9,2,1,1] => [5,4,3,1] => 12600
[9,1,1,1,1] => [6,4,3] => 1008
[8,5] => [5,3,3,2] => 18900
[8,4,1] => [5,3,3,1,1] => 151200
[8,3,2] => [4,4,1,1,1,1,1] => 16450560
[8,3,1,1] => [4,4,2,1,1,1] => 1827840
[8,2,2,1] => [5,2,2,2,2] => 268800
[8,2,1,1,1] => [6,3,3,1] => 11340
[8,1,1,1,1,1] => [7,3,3] => 882
[7,6] => [3,3,3,2,2] => 362880
[7,5,1] => [3,3,2,2,2,1] => 4112640
[7,4,2] => [5,3,1,1,1,1,1] => 15422400
[7,4,1,1] => [5,2,2,2,1,1] => 2284800
[7,3,3] => [5,3,2,1,1,1] => 1713600
[7,3,2,1] => [4,2,2,1,1,1,1,1] => 312560640
[7,3,1,1,1] => [6,3,2,1,1] => 120960
[7,2,2,2] => [4,3,3,2,1] => 241920
[7,2,2,1,1] => [6,4,2,1] => 10080
[7,2,1,1,1,1] => [3,3,3,3,1] => 272160
[7,1,1,1,1,1,1] => [4,3,3,3] => 22680
[6,6,1] => [5,3,2,2,1] => 201600
[6,5,2] => [3,3,2,2,1,1,1] => 37013760
[6,5,1,1] => [4,3,2,2,1,1] => 2741760
[6,4,1,1,1] => [2,2,2,2,2,1,1,1] => 625121280
[6,3,2,2] => [4,3,1,1,1,1,1,1] => 234420480
[6,3,2,1,1] => [4,3,2,1,1,1,1] => 24675840
[6,3,1,1,1,1] => [3,3,3,2,1,1] => 3084480
[6,2,2,2,1] => [2,2,2,2,2,2,1] => 65802240
[6,2,2,1,1,1] => [4,4,2,2,1] => 215040
[6,2,1,1,1,1,1] => [7,3,2,1] => 8820
[6,1,1,1,1,1,1,1] => [8,3,2] => 672
[5,5,3] => [3,2,2,2,1,1,1,1] => 468840960
[5,5,2,1] => [5,1,1,1,1,1,1,1,1] => 1953504000
[5,5,1,1,1] => [4,2,2,2,1,1,1] => 32901120
[5,4,4] => [5,2,2,1,1,1,1] => 20563200
[5,4,1,1,1,1] => [6,2,2,1,1,1] => 1370880
[5,3,2,1,1,1] => [6,1,1,1,1,1,1,1] => 117210240
[5,3,1,1,1,1,1] => [7,2,1,1,1,1] => 799680
[5,2,2,2,2] => [4,2,2,2,2,1] => 3655680
[5,2,2,2,1,1] => [6,2,2,2,1] => 161280
[5,2,2,1,1,1,1] => [7,2,2,1,1] => 94080
[5,2,1,1,1,1,1,1] => [8,3,1,1] => 5040
[5,1,1,1,1,1,1,1,1] => [9,2,2] => 504
[4,4,4,1] => [5,2,1,1,1,1,1,1] => 195350400
[4,4,2,2,1] => [3,3,2,1,1,1,1,1] => 351630720
[4,4,2,1,1,1] => [6,2,1,1,1,1,1] => 12337920
[4,4,1,1,1,1,1] => [7,3,1,1,1] => 70560
[4,3,3,3] => [3,3,3,1,1,1,1] => 27760320
[4,3,3,2,1] => [3,2,2,2,2,1,1] => 49351680
[4,3,3,1,1,1] => [6,3,1,1,1,1] => 1028160
[4,3,2,2,2] => [4,3,3,1,1,1] => 2056320
[4,3,2,2,1,1] => [6,4,1,1,1] => 80640
[4,3,2,1,1,1,1] => [7,1,1,1,1,1,1] => 7197120
[4,3,1,1,1,1,1,1] => [8,1,1,1,1,1] => 456960
[4,2,2,2,2,1] => [4,4,3,1,1] => 161280
[4,2,2,2,1,1,1] => [7,4,1,1] => 5880
[4,2,2,1,1,1,1,1] => [8,2,1,1,1] => 53760
[4,2,1,1,1,1,1,1,1] => [9,1,1,1,1] => 30240
[4,1,1,1,1,1,1,1,1,1] => [10,2,1] => 280
[3,3,3,3,1] => [3,2,2,2,2,2] => 5483520
[3,3,3,2,2] => [4,3,2,2,2] => 322560
[3,3,3,2,1,1] => [6,3,2,2] => 15120
[3,3,3,1,1,1,1] => [7,2,2,2] => 11760
[3,3,2,2,2,1] => [4,4,3,2] => 20160
[3,3,2,2,1,1,1] => [7,4,2] => 784
[3,3,2,1,1,1,1,1] => [8,2,2,1] => 6720
[3,3,1,1,1,1,1,1,1] => [9,2,1,1] => 3780
[3,2,2,2,2,2] => [4,4,4,1] => 13440
[3,2,2,2,2,1,1] => [7,5,1] => 490
[3,2,2,2,1,1,1,1] => [8,4,1] => 448
[3,2,2,1,1,1,1,1,1] => [9,3,1] => 378
[3,2,1,1,1,1,1,1,1,1] => [10,1,1,1] => 2100
[3,1,1,1,1,1,1,1,1,1,1] => [11,1,1] => 154
[2,2,2,2,2,2,1] => [5,4,4] => 1120
[2,2,2,2,2,1,1,1] => [8,5] => 40
[2,2,2,2,1,1,1,1,1] => [9,4] => 36
[2,2,2,1,1,1,1,1,1,1] => [10,3] => 30
[2,2,1,1,1,1,1,1,1,1,1] => [11,2] => 22
[2,1,1,1,1,1,1,1,1,1,1,1] => [12,1] => 12
[1,1,1,1,1,1,1,1,1,1,1,1,1] => [13] => 1
[14] => [7,7] => 49
[13,1] => [7,6,1] => 630
[12,2] => [6,6,1,1] => 8640
[12,1,1] => [6,6,2] => 1080
[11,3] => [6,5,2,1] => 14400
[11,2,1] => [6,5,1,1,1] => 122400
[11,1,1,1] => [6,5,3] => 1350
[10,4] => [5,5,2,2] => 24000
[10,3,1] => [5,5,1,1,1,1] => 1836000
[10,2,2] => [5,5,2,1,1] => 204000
[10,2,1,1] => [5,5,3,1] => 18000
[10,1,1,1,1] => [6,4,4] => 1440
[9,5] => [5,4,3,2] => 28800
[9,4,1] => [5,4,3,1,1] => 244800
[9,3,2] => [5,4,1,1,1,1,1] => 27907200
[9,3,1,1] => [5,4,2,1,1,1] => 2937600
[9,2,2,1] => [5,4,2,2,1] => 326400
[9,2,1,1,1] => [6,4,3,1] => 17280
[9,1,1,1,1,1] => [7,4,3] => 1260
[8,6] => [3,3,3,3,2] => 660960
[8,5,1] => [5,3,3,2,1] => 367200
[8,4,2] => [5,3,2,1,1,1,1] => 41860800
[8,4,1,1] => [5,3,2,2,1,1] => 4406400
[8,3,3] => [5,3,3,1,1,1] => 3304800
[8,3,2,1] => [4,4,1,1,1,1,1,1] => 446515200
[8,3,1,1,1] => [6,3,3,1,1] => 220320
[8,2,2,2] => [4,4,2,2,2] => 522240
[8,2,2,1,1] => [6,4,2,2] => 23040
[8,2,1,1,1,1] => [7,3,3,1] => 15120
[8,1,1,1,1,1,1] => [4,4,3,3] => 34560
[7,7] => [6,3,3,2] => 25920
[7,6,1] => [3,3,3,2,2,1] => 7931520
[7,5,2] => [3,3,2,2,2,1,1] => 100465920
[7,5,1,1] => [4,4,2,2,1,1] => 4700160
[7,4,3] => [5,2,2,2,1,1,1] => 55814400
[7,4,1,1,1] => [6,2,2,2,1,1] => 3525120
[7,3,3,1] => [5,3,1,1,1,1,1,1] => 418608000
[7,3,2,2] => [4,3,3,1,1,1,1] => 50232960
[7,3,2,1,1] => [6,4,1,1,1,1] => 1762560
[7,3,1,1,1,1] => [3,3,3,3,1,1] => 5948640
[7,2,2,2,1] => [6,2,2,2,2] => 391680
[7,2,2,1,1,1] => [4,3,3,2,2] => 587520
[7,2,1,1,1,1,1] => [4,3,3,3,1] => 440640
[7,1,1,1,1,1,1,1] => [8,3,3] => 1080
[6,6,2] => [4,3,3,2,1,1] => 5287680
[6,6,1,1] => [6,4,2,1,1] => 195840
[6,5,2,1] => [4,3,2,1,1,1,1,1] => 669772800
[6,5,1,1,1] => [2,2,2,2,2,2,1,1] => 1786060800
[6,4,4] => [3,2,2,2,2,1,1,1] => 1339545600
[6,4,1,1,1,1] => [4,2,2,2,2,1,1] => 89303040
[6,3,2,2,1] => [3,3,3,1,1,1,1,1] => 753494400
[6,3,2,1,1,1] => [4,4,2,1,1,1,1] => 44651520
[6,3,1,1,1,1,1] => [7,3,2,1,1] => 171360
[6,2,2,2,2] => [2,2,2,2,2,2,2] => 178606080
[6,2,2,2,1,1] => [4,2,2,2,2,2] => 9400320
[6,2,2,1,1,1,1] => [7,4,2,1] => 13440
[6,2,1,1,1,1,1,1] => [8,3,2,1] => 11520
[6,1,1,1,1,1,1,1,1] => [9,3,2] => 810
[5,5,4] => [4,3,2,2,1,1,1] => 66977280
[5,5,1,1,1,1] => [5,2,2,2,2,1] => 5875200
[5,4,1,1,1,1,1] => [7,2,2,1,1,1] => 2056320
[5,3,3,1,1,1] => [6,2,1,1,1,1,1,1] => 334886400
[5,3,2,2,2] => [3,3,2,2,1,1,1,1] => 1004659200
[5,3,2,2,1,1] => [6,3,1,1,1,1,1] => 25116480
[5,3,2,1,1,1,1] => [7,1,1,1,1,1,1,1] => 195350400
[5,3,1,1,1,1,1,1] => [8,2,1,1,1,1] => 1175040
[5,2,2,2,2,1] => [4,4,3,2,1] => 391680
[5,2,2,2,1,1,1] => [7,2,2,2,1] => 228480
[5,2,2,1,1,1,1,1] => [8,2,2,1,1] => 130560
[5,2,1,1,1,1,1,1,1] => [9,3,1,1] => 6480
[5,1,1,1,1,1,1,1,1,1] => [10,2,2] => 600
[4,4,4,1,1] => [4,2,2,2,1,1,1,1] => 893030400
[4,4,3,3] => [5,2,2,1,1,1,1,1] => 558144000
[4,4,3,1,1,1] => [6,2,2,1,1,1,1] => 33488640
[4,4,2,2,2] => [3,3,3,2,1,1,1] => 75349440
[4,4,2,2,1,1] => [6,3,2,1,1,1] => 2643840
[4,4,2,1,1,1,1] => [7,2,1,1,1,1,1] => 19535040
[4,4,1,1,1,1,1,1] => [8,3,1,1,1] => 97920
[4,3,3,3,1] => [3,2,2,2,2,2,1] => 133954560
[4,3,3,2,2] => [4,3,2,2,2,1] => 7050240
[4,3,3,2,1,1] => [6,3,2,2,1] => 293760
[4,3,3,1,1,1,1] => [7,3,1,1,1,1] => 1542240
[4,3,2,2,2,1] => [4,4,3,1,1,1] => 3525120
[4,3,2,2,1,1,1] => [7,4,1,1,1] => 114240
[4,3,2,1,1,1,1,1] => [8,1,1,1,1,1,1] => 11162880
[4,3,1,1,1,1,1,1,1] => [9,1,1,1,1,1] => 660960
[4,2,2,2,2,2] => [4,4,4,1,1] => 261120
[4,2,2,2,2,1,1] => [7,5,1,1] => 8400
[4,2,2,2,1,1,1,1] => [8,4,1,1] => 7680
[4,2,2,1,1,1,1,1,1] => [9,2,1,1,1] => 73440
[4,2,1,1,1,1,1,1,1,1] => [10,1,1,1,1] => 40800
[4,1,1,1,1,1,1,1,1,1,1] => [11,2,1] => 330
[3,3,3,3,2] => [3,3,2,2,2,2] => 10575360
[3,3,3,3,1,1] => [5,3,2,2,2] => 489600
[3,3,3,2,2,1] => [5,3,3,3] => 32400
[3,3,3,2,1,1,1] => [7,3,2,2] => 20160
[3,3,3,1,1,1,1,1] => [8,2,2,2] => 15360
[3,3,2,2,2,2] => [4,4,4,2] => 30720
[3,3,2,2,2,1,1] => [7,5,2] => 1050
[3,3,2,2,1,1,1,1] => [8,4,2] => 960
[3,3,2,1,1,1,1,1,1] => [9,2,2,1] => 8640
[3,3,1,1,1,1,1,1,1,1] => [10,2,1,1] => 4800
[3,2,2,2,2,2,1] => [5,4,4,1] => 19200
[3,2,2,2,2,1,1,1] => [8,5,1] => 600
[3,2,2,2,1,1,1,1,1] => [9,4,1] => 540
[3,2,2,1,1,1,1,1,1,1] => [10,3,1] => 450
[3,2,1,1,1,1,1,1,1,1,1] => [11,1,1,1] => 2640
[3,1,1,1,1,1,1,1,1,1,1,1] => [12,1,1] => 180
[2,2,2,2,2,2,2] => [5,5,4] => 1500
[2,2,2,2,2,2,1,1] => [8,6] => 48
[2,2,2,2,2,1,1,1,1] => [9,5] => 45
[2,2,2,2,1,1,1,1,1,1] => [10,4] => 40
[2,2,2,1,1,1,1,1,1,1,1] => [11,3] => 33
[2,2,1,1,1,1,1,1,1,1,1,1] => [12,2] => 24
[2,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1] => 13
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14] => 1
[15] => [8,7] => 56
[14,1] => [7,7,1] => 784
[13,2] => [7,6,1,1] => 11424
[13,1,1] => [7,6,2] => 1344
[12,3] => [6,6,2,1] => 19584
[12,2,1] => [6,6,1,1,1] => 176256
[12,1,1,1] => [6,6,3] => 1728
[11,4] => [6,5,2,2] => 32640
[11,3,1] => [6,5,1,1,1,1] => 2790720
[11,2,2] => [6,5,2,1,1] => 293760
[11,2,1,1] => [6,5,3,1] => 24480
[11,1,1,1,1] => [6,5,4] => 1920
[10,5] => [5,5,3,2] => 40800
[10,4,1] => [5,5,3,1,1] => 367200
[10,3,2] => [5,5,1,1,1,1,1] => 46512000
[10,3,1,1] => [5,5,2,1,1,1] => 4651200
[10,2,2,1] => [5,5,2,2,1] => 489600
[10,2,1,1,1] => [6,4,4,1] => 26112
[10,1,1,1,1,1] => [7,4,4] => 1792
[9,6] => [6,3,3,3] => 44064
[9,5,1] => [5,4,3,2,1] => 587520
[9,4,2] => [5,4,2,1,1,1,1] => 74419200
[9,4,1,1] => [5,4,2,2,1,1] => 7441920
[9,3,3] => [5,4,3,1,1,1] => 5581440
[9,3,2,1] => [5,4,1,1,1,1,1,1] => 781401600
[9,3,1,1,1] => [6,4,3,1,1] => 352512
[9,2,2,2] => [5,4,2,2,2] => 783360
[9,2,2,1,1] => [6,4,3,2] => 39168
[9,2,1,1,1,1] => [7,4,3,1] => 22848
[9,1,1,1,1,1,1] => [8,4,3] => 1536
[8,7] => [3,3,3,3,3] => 1189728
[8,6,1] => [3,3,3,3,2,1] => 15069888
[8,5,2] => [5,3,3,2,1,1] => 8372160
[8,5,1,1] => [6,4,2,2,1] => 470016
[8,4,3] => [5,3,2,2,1,1,1] => 111628800
[8,4,1,1,1] => [4,4,2,2,2,1] => 11907072
[8,3,3,1] => [5,3,2,1,1,1,1,1] => 1172102400
[8,3,2,2] => [4,3,3,2,1,1,1] => 133954560
[8,3,2,1,1] => [6,4,2,1,1,1] => 4465152
[8,3,1,1,1,1] => [7,3,3,1,1] => 308448
[8,2,2,2,1] => [6,3,2,2,2] => 705024
[8,2,2,1,1,1] => [7,4,2,2] => 30464
[8,2,1,1,1,1,1] => [4,4,3,3,1] => 705024
[8,1,1,1,1,1,1,1] => [4,4,4,3] => 52224
[7,7,1] => [6,3,3,2,1] => 528768
[7,6,2] => [3,3,3,2,2,1,1] => 200931840
[7,6,1,1] => [4,3,2,2,2,2] => 17860608
[7,5,2,1] => [4,4,2,1,1,1,1,1] => 1250242560
[7,5,1,1,1] => [4,2,2,2,2,2,1] => 238141440
[7,4,4] => [4,3,2,2,2,1,1] => 178606080
[7,4,2,1,1] => [6,3,1,1,1,1,1,1] => 703261440
[7,4,1,1,1,1] => [4,3,3,2,2,1] => 13395456
[7,3,3,2] => [3,3,3,2,1,1,1,1] => 2109784320
[7,3,3,1,1] => [6,3,2,1,1,1,1] => 66977280
[7,3,2,2,1] => [6,3,3,1,1,1] => 5023296
[7,3,2,1,1,1] => [3,3,3,3,1,1,1] => 150698880
[7,3,1,1,1,1,1] => [4,3,3,3,1,1] => 10046592
[7,2,2,2,2] => [5,2,2,2,2,2] => 14883840
[7,2,2,2,1,1] => [4,4,3,2,2] => 940032
[7,2,2,1,1,1,1] => [4,3,3,3,2] => 1057536
[7,2,1,1,1,1,1,1] => [8,3,3,1] => 19584
[7,1,1,1,1,1,1,1,1] => [9,3,3] => 1296
[6,6,3] => [5,2,2,2,2,1,1] => 148838400
[6,6,2,1] => [6,4,1,1,1,1,1] => 44651520
[6,6,1,1,1] => [6,2,2,2,2,1] => 8930304
[6,4,1,1,1,1,1] => [7,2,2,2,1,1] => 5209344
[6,3,2,2,1,1] => [5,3,3,1,1,1,1] => 83721600
[6,3,2,1,1,1,1] => [7,4,1,1,1,1] => 2604672
[6,3,1,1,1,1,1,1] => [8,3,2,1,1] => 235008
[6,2,2,2,2,1] => [3,2,2,2,2,2,2] => 357212160
[6,2,2,2,1,1,1] => [7,2,2,2,2] => 548352
[6,2,2,1,1,1,1,1] => [8,4,2,1] => 17408
[6,2,1,1,1,1,1,1,1] => [9,3,2,1] => 14688
[6,1,1,1,1,1,1,1,1,1] => [10,3,2] => 960
[5,5,5] => [6,3,2,2,1,1] => 6697728
[5,5,2,1,1,1] => [4,3,3,1,1,1,1,1] => 1406522880
[5,5,1,1,1,1,1] => [7,4,2,1,1] => 274176
[5,4,1,1,1,1,1,1] => [8,2,2,1,1,1] => 2976768
[5,3,3,2,1,1] => [6,2,2,1,1,1,1,1] => 937681920
[5,3,3,1,1,1,1] => [7,2,1,1,1,1,1,1] => 546981120
[5,3,2,2,2,1] => [4,4,3,1,1,1,1] => 89303040
[5,3,2,2,1,1,1] => [7,3,1,1,1,1,1] => 39070080
[5,3,2,1,1,1,1,1] => [8,1,1,1,1,1,1,1] => 312560640
[5,3,1,1,1,1,1,1,1] => [9,2,1,1,1,1] => 1674432
[5,2,2,2,2,2] => [4,4,4,2,1] => 626688
[5,2,2,2,2,1,1] => [7,5,2,1] => 19040
[5,2,2,2,1,1,1,1] => [8,2,2,2,1] => 313344
[5,2,2,1,1,1,1,1,1] => [9,2,2,1,1] => 176256
[5,2,1,1,1,1,1,1,1,1] => [10,3,1,1] => 8160
[5,1,1,1,1,1,1,1,1,1,1] => [11,2,2] => 704
[4,4,4,3] => [4,3,2,2,1,1,1,1] => 1875363840
[4,4,4,1,1,1] => [4,4,2,2,1,1,1] => 119070720
[4,4,3,2,2] => [5,2,2,2,1,1,1,1] => 1562803200
[4,4,3,2,1,1] => [6,2,2,2,1,1,1] => 89303040
[4,4,3,1,1,1,1] => [7,2,2,1,1,1,1] => 52093440
[4,4,2,2,2,1] => [4,4,3,2,1,1] => 8930304
[4,4,2,2,1,1,1] => [7,3,2,1,1,1] => 3907008
[4,4,2,1,1,1,1,1] => [8,2,1,1,1,1,1] => 29767680
[4,4,1,1,1,1,1,1,1] => [9,3,1,1,1] => 132192
[4,3,3,3,2] => [3,3,2,2,2,2,1] => 267909120
[4,3,3,3,1,1] => [5,3,2,2,2,1] => 11162880
[4,3,3,2,2,1] => [5,3,3,3,1] => 660960
[4,3,3,2,1,1,1] => [7,3,2,2,1] => 411264
[4,3,3,1,1,1,1,1] => [8,3,1,1,1,1] => 2232576
[4,3,2,2,2,2] => [4,4,4,1,1,1] => 5953536
[4,3,2,2,2,1,1] => [7,5,1,1,1] => 171360
[4,3,2,2,1,1,1,1] => [8,4,1,1,1] => 156672
[4,3,2,1,1,1,1,1,1] => [9,1,1,1,1,1,1] => 16744320
[4,3,1,1,1,1,1,1,1,1] => [10,1,1,1,1,1] => 930240
[4,2,2,2,2,2,1] => [5,4,4,1,1] => 391680
[4,2,2,2,2,1,1,1] => [8,5,1,1] => 10880
[4,2,2,2,1,1,1,1,1] => [9,4,1,1] => 9792
[4,2,2,1,1,1,1,1,1,1] => [10,2,1,1,1] => 97920
[4,2,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1] => 53856
[4,1,1,1,1,1,1,1,1,1,1,1] => [12,2,1] => 384
[3,3,3,3,3] => [5,3,3,2,2] => 881280
[3,3,3,3,2,1] => [3,3,3,2,2,2] => 20093184
[3,3,3,3,1,1,1] => [7,3,3,2] => 34272
[3,3,3,2,2,2] => [5,4,3,3] => 48960
[3,3,3,2,2,1,1] => [7,5,3] => 1680
[3,3,3,2,1,1,1,1] => [8,3,2,2] => 26112
[3,3,3,1,1,1,1,1,1] => [9,2,2,2] => 19584
[3,3,2,2,2,2,1] => [5,4,4,2] => 43520
[3,3,2,2,2,1,1,1] => [8,5,2] => 1280
[3,3,2,2,1,1,1,1,1] => [9,4,2] => 1152
[3,3,2,1,1,1,1,1,1,1] => [10,2,2,1] => 10880
[3,3,1,1,1,1,1,1,1,1,1] => [11,2,1,1] => 5984
[3,2,2,2,2,2,2] => [5,5,4,1] => 27200
[3,2,2,2,2,2,1,1] => [8,6,1] => 768
[3,2,2,2,2,1,1,1,1] => [9,5,1] => 720
[3,2,2,2,1,1,1,1,1,1] => [10,4,1] => 640
[3,2,2,1,1,1,1,1,1,1,1] => [11,3,1] => 528
[3,2,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1] => 3264
[3,1,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1] => 208
[2,2,2,2,2,2,2,1] => [5,5,5] => 2000
[2,2,2,2,2,2,1,1,1] => [9,6] => 54
[2,2,2,2,2,1,1,1,1,1] => [10,5] => 50
[2,2,2,2,1,1,1,1,1,1,1] => [11,4] => 44
[2,2,2,1,1,1,1,1,1,1,1,1] => [12,3] => 36
[2,2,1,1,1,1,1,1,1,1,1,1,1] => [13,2] => 26
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1] => 14
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15] => 1
[16] => [8,8] => 64
[15,1] => [8,7,1] => 952
[14,2] => [7,7,1,1] => 14994
[14,1,1] => [7,7,2] => 1666
[13,3] => [7,6,2,1] => 25704
[13,2,1] => [7,6,1,1,1] => 244188
[13,1,1,1] => [7,6,3] => 2142
[12,4] => [6,6,2,2] => 44064
[12,3,1] => [6,6,1,1,1,1] => 4186080
[12,2,2] => [6,6,2,1,1] => 418608
[12,2,1,1] => [6,6,3,1] => 33048
[12,1,1,1,1] => [6,6,4] => 2448
[11,5] => [6,5,3,2] => 55080
[11,4,1] => [6,5,3,1,1] => 523260
[11,3,2] => [6,5,1,1,1,1,1] => 73256400
[11,3,1,1] => [6,5,2,1,1,1] => 6976800
[11,2,2,1] => [6,5,2,2,1] => 697680
[11,2,1,1,1] => [6,5,4,1] => 36720
[11,1,1,1,1,1] => [7,5,4] => 2380
[10,6] => [5,5,3,3] => 68850
[10,5,1] => [5,5,3,2,1] => 872100
[10,4,2] => [5,5,2,1,1,1,1] => 122094000
[10,4,1,1] => [5,5,2,2,1,1] => 11628000
[10,3,3] => [5,5,3,1,1,1] => 8721000
[10,3,2,1] => [5,5,1,1,1,1,1,1] => 1343034000
[10,3,1,1,1] => [6,4,4,1,1] => 558144
[10,2,2,2] => [5,5,2,2,2] => 1162800
[10,2,2,1,1] => [6,4,4,2] => 58752
[10,2,1,1,1,1] => [7,4,4,1] => 34272
[10,1,1,1,1,1,1] => [8,4,4] => 2176
[9,7] => [4,3,3,3,3] => 1883736
[9,6,1] => [6,3,3,3,1] => 941868
[9,5,2] => [5,4,3,2,1,1] => 13953600
[9,5,1,1] => [6,4,3,2,1] => 837216
[9,4,3] => [5,4,2,2,1,1,1] => 195350400
[9,4,1,1,1] => [5,4,2,2,2,1] => 18604800
[9,3,2,2] => [5,4,3,1,1,1,1] => 146512800
[9,3,2,1,1] => [6,4,3,1,1,1] => 8372160
[9,3,1,1,1,1] => [7,4,3,1,1] => 488376
[9,2,2,2,1] => [6,3,3,2,2] => 1255824
[9,2,2,1,1,1] => [7,4,3,2] => 51408
[9,2,1,1,1,1,1] => [8,4,3,1] => 29376
[9,1,1,1,1,1,1,1] => [4,4,4,4] => 78336
[8,8] => [6,4,3,3] => 66096
[8,7,1] => [3,3,3,3,3,1] => 28256040
[8,6,2] => [3,3,3,3,2,1,1] => 395584560
[8,6,1,1] => [4,3,3,2,2,2] => 33488640
[8,5,3] => [4,3,3,2,2,1,1] => 351630720
[8,5,2,1] => [6,4,2,1,1,1,1] => 117210240
[8,5,1,1,1] => [6,2,2,2,2,2] => 22325760
[8,4,4] => [6,4,2,2,1,1] => 11162880
[8,4,1,1,1,1] => [7,4,2,2,1] => 651168
[8,3,3,1,1] => [6,3,2,2,1,1,1] => 175815360
[8,3,2,2,1] => [6,3,2,2,2,1] => 16744320
[8,3,2,1,1,1] => [7,4,2,1,1,1] => 6511680
[8,3,1,1,1,1,1] => [4,4,3,3,1,1] => 16744320
[8,2,2,2,2] => [4,4,4,2,2] => 1488384
[8,2,2,2,1,1] => [7,5,2,2] => 42840
[8,2,2,1,1,1,1] => [4,4,3,3,2] => 1674432
[8,2,1,1,1,1,1,1] => [4,4,4,3,1] => 1116288
[8,1,1,1,1,1,1,1,1] => [9,4,3] => 1836
[7,7,2] => [6,3,3,2,1,1] => 12558240
[7,7,1,1] => [6,4,2,2,2] => 1116288
[7,6,3] => [4,4,2,2,2,1,1] => 312560640
[7,6,1,1,1] => [4,4,2,2,2,2] => 29767680
[7,4,1,1,1,1,1] => [4,3,3,3,2,1] => 25116480
[7,3,3,1,1,1] => [5,3,3,2,1,1,1] => 219769200
[7,3,2,2,2] => [5,3,2,2,2,1,1] => 293025600
[7,3,2,2,1,1] => [5,3,3,3,1,1] => 15697800
[7,3,2,1,1,1,1] => [4,3,3,3,1,1,1] => 263723040
[7,3,1,1,1,1,1,1] => [8,3,3,1,1] => 418608
[7,2,2,2,2,1] => [3,3,2,2,2,2,2] => 703261440
[7,2,2,2,1,1,1] => [5,4,3,2,2] => 1395360
[7,2,2,1,1,1,1,1] => [8,4,2,2] => 39168
[7,2,1,1,1,1,1,1,1] => [9,3,3,1] => 24786
[7,1,1,1,1,1,1,1,1,1] => [10,3,3] => 1530
[6,6,2,2] => [6,3,2,1,1,1,1,1] => 1933968960
[6,6,2,1,1] => [6,3,3,1,1,1,1] => 131861520
[6,6,1,1,1,1] => [4,2,2,2,2,2,2] => 625121280
[6,5,5] => [6,2,2,2,2,1,1] => 234420480
[6,5,1,1,1,1,1] => [7,2,2,2,2,1] => 13023360
[6,4,2,1,1,1,1] => [7,3,1,1,1,1,1,1] => 1128148560
[6,4,1,1,1,1,1,1] => [8,2,2,2,1,1] => 7441920
[6,3,3,1,1,1,1] => [7,3,2,1,1,1,1] => 102558960
[6,3,2,2,1,1,1] => [7,3,3,1,1,1] => 7325640
[6,3,2,1,1,1,1,1] => [8,4,1,1,1,1] => 3720960
[6,3,1,1,1,1,1,1,1] => [9,3,2,1,1] => 313956
[6,2,2,2,2,2] => [5,3,2,2,2,2] => 27907200
[6,2,2,2,2,1,1] => [7,3,2,2,2] => 976752
[6,2,2,2,1,1,1,1] => [8,2,2,2,2] => 744192
[6,2,2,1,1,1,1,1,1] => [9,4,2,1] => 22032
[6,2,1,1,1,1,1,1,1,1] => [10,3,2,1] => 18360
[6,1,1,1,1,1,1,1,1,1,1] => [11,3,2] => 1122
[5,5,5,1] => [6,4,1,1,1,1,1,1] => 1289312640
[5,5,2,1,1,1,1] => [7,4,1,1,1,1,1] => 68372640
[5,5,1,1,1,1,1,1] => [8,4,2,1,1] => 372096
[5,4,1,1,1,1,1,1,1] => [9,2,2,1,1,1] => 4186080
[5,3,3,2,2,1] => [4,4,3,2,1,1,1] => 234420480
[5,3,3,2,1,1,1] => [7,2,2,1,1,1,1,1] => 1504198080
[5,3,3,1,1,1,1,1] => [8,2,1,1,1,1,1,1] => 859541760
[5,3,2,2,2,2] => [4,4,4,1,1,1,1] => 156280320
[5,3,2,2,2,1,1] => [7,5,1,1,1,1] => 4069800
[5,3,2,2,1,1,1,1] => [8,3,1,1,1,1,1] => 58605120
[5,3,2,1,1,1,1,1,1] => [9,1,1,1,1,1,1,1] => 483492240
[5,3,1,1,1,1,1,1,1,1] => [10,2,1,1,1,1] => 2325600
[5,2,2,2,2,2,1] => [5,4,4,2,1] => 930240
[5,2,2,2,2,1,1,1] => [8,5,2,1] => 24480
[5,2,2,2,1,1,1,1,1] => [9,2,2,2,1] => 418608
[5,2,2,1,1,1,1,1,1,1] => [10,2,2,1,1] => 232560
[5,2,1,1,1,1,1,1,1,1,1] => [11,3,1,1] => 10098
[5,1,1,1,1,1,1,1,1,1,1,1] => [12,2,2] => 816
[4,4,4,4] => [4,3,2,2,2,2,1] => 468840960
[4,4,4,1,1,1,1] => [7,3,2,2,1,1] => 9767520
[4,4,3,3,1,1] => [5,2,2,2,2,2,1] => 390700800
[4,4,3,2,2,1] => [4,4,3,2,2,1] => 22325760
[4,4,3,2,1,1,1] => [7,2,2,2,1,1,1] => 136745280
[4,4,3,1,1,1,1,1] => [8,2,2,1,1,1,1] => 78140160
[4,4,2,2,2,2] => [4,4,4,2,1,1] => 14883840
[4,4,2,2,2,1,1] => [7,5,2,1,1] => 406980
[4,4,2,2,1,1,1,1] => [8,3,2,1,1,1] => 5581440
[4,4,2,1,1,1,1,1,1] => [9,2,1,1,1,1,1] => 43953840
[4,4,1,1,1,1,1,1,1,1] => [10,3,1,1,1] => 174420
[4,3,3,3,3] => [5,3,3,2,2,1] => 20930400
[4,3,3,3,2,1] => [3,3,3,2,2,2,1] => 527446080
[4,3,3,3,1,1,1] => [7,3,3,2,1] => 732564
[4,3,3,2,2,2] => [5,4,3,3,1] => 1046520
[4,3,3,2,2,1,1] => [7,5,3,1] => 32130
[4,3,3,2,1,1,1,1] => [8,3,2,2,1] => 558144
[4,3,3,1,1,1,1,1,1] => [9,3,1,1,1,1] => 3139560
[4,3,2,2,2,2,1] => [5,4,4,1,1,1] => 9302400
[4,3,2,2,2,1,1,1] => [8,5,1,1,1] => 232560
[4,3,2,2,1,1,1,1,1] => [9,4,1,1,1] => 209304
[4,3,2,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1] => 24418800
[4,3,1,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1] => 1279080
[4,2,2,2,2,2,2] => [5,5,4,1,1] => 581400
[4,2,2,2,2,2,1,1] => [8,6,1,1] => 14688
[4,2,2,2,2,1,1,1,1] => [9,5,1,1] => 13770
[4,2,2,2,1,1,1,1,1,1] => [10,4,1,1] => 12240
[4,2,2,1,1,1,1,1,1,1,1] => [11,2,1,1,1] => 127908
[4,2,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1] => 69768
[4,1,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1] => 442
[3,3,3,3,3,1] => [3,3,3,3,2,2] => 37674720
[3,3,3,3,2,2] => [5,3,3,3,2] => 1569780
[3,3,3,3,2,1,1] => [7,3,3,3] => 57834
[3,3,3,3,1,1,1,1] => [8,3,3,2] => 44064
[3,3,3,2,2,2,1] => [5,4,4,3] => 73440
[3,3,3,2,2,1,1,1] => [8,5,3] => 2040
[3,3,3,2,1,1,1,1,1] => [9,3,2,2] => 33048
[3,3,3,1,1,1,1,1,1,1] => [10,2,2,2] => 24480
[3,3,2,2,2,2,2] => [5,5,4,2] => 61200
[3,3,2,2,2,2,1,1] => [8,6,2] => 1632
[3,3,2,2,2,1,1,1,1] => [9,5,2] => 1530
[3,3,2,2,1,1,1,1,1,1] => [10,4,2] => 1360
[3,3,2,1,1,1,1,1,1,1,1] => [11,2,2,1] => 13464
[3,3,1,1,1,1,1,1,1,1,1,1] => [12,2,1,1] => 7344
[3,2,2,2,2,2,2,1] => [5,5,5,1] => 38250
[3,2,2,2,2,2,1,1,1] => [9,6,1] => 918
[3,2,2,2,2,1,1,1,1,1] => [10,5,1] => 850
[3,2,2,2,1,1,1,1,1,1,1] => [11,4,1] => 748
[3,2,2,1,1,1,1,1,1,1,1,1] => [12,3,1] => 612
[3,2,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1] => 3978
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1] => 238
[2,2,2,2,2,2,2,2] => [6,5,5] => 2550
[2,2,2,2,2,2,2,1,1] => [9,7] => 63
[2,2,2,2,2,2,1,1,1,1] => [10,6] => 60
[2,2,2,2,2,1,1,1,1,1,1] => [11,5] => 55
[2,2,2,2,1,1,1,1,1,1,1,1] => [12,4] => 48
[2,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3] => 39
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2] => 28
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1] => 15
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16] => 1
[17] => [9,8] => 72
[16,1] => [8,8,1] => 1152
[15,2] => [8,7,1,1] => 19152
[15,1,1] => [8,7,2] => 2016
[14,3] => [7,7,2,1] => 33516
[14,2,1] => [7,7,1,1,1] => 335160
[14,1,1,1] => [7,7,3] => 2646
[13,4] => [7,6,2,2] => 57456
[13,3,1] => [7,6,1,1,1,1] => 6032880
[13,2,2] => [7,6,2,1,1] => 574560
[13,2,1,1] => [7,6,3,1] => 43092
[13,1,1,1,1] => [7,6,4] => 3024
[12,5] => [6,6,3,2] => 73872
[12,4,1] => [6,6,3,1,1] => 738720
[12,3,2] => [6,6,1,1,1,1,1] => 113762880
[12,3,1,1] => [6,6,2,1,1,1] => 10342080
[12,2,2,1] => [6,6,2,2,1] => 984960
[12,2,1,1,1] => [6,6,4,1] => 49248
[12,1,1,1,1,1] => [7,5,5] => 3150
[11,6] => [6,5,3,3] => 92340
[11,5,1] => [6,5,3,2,1] => 1231200
[11,4,2] => [6,5,2,1,1,1,1] => 189604800
[11,4,1,1] => [6,5,2,2,1,1] => 17236800
[11,3,3] => [6,5,3,1,1,1] => 12927600
[11,3,1,1,1] => [6,5,4,1,1] => 820800
[11,2,2,2] => [6,5,2,2,2] => 1641600
[11,2,2,1,1] => [6,5,4,2] => 82080
[11,2,1,1,1,1] => [7,5,4,1] => 47880
[11,1,1,1,1,1,1] => [8,5,4] => 2880
[10,7] => [7,4,3,3] => 86184
[10,6,1] => [5,5,3,3,1] => 1539000
[10,5,2] => [5,5,3,2,1,1] => 21546000
[10,5,1,1] => [6,4,4,2,1] => 1313280
[10,4,3] => [5,5,2,2,1,1,1] => 316008000
[10,4,1,1,1] => [5,5,2,2,2,1] => 28728000
[10,3,2,2] => [5,5,3,1,1,1,1] => 237006000
[10,3,2,1,1] => [6,4,4,1,1,1] => 13789440
[10,3,1,1,1,1] => [7,4,4,1,1] => 766080
[10,2,2,2,1] => [6,3,3,3,2] => 2216160
[10,2,2,1,1,1] => [7,4,4,2] => 76608
[10,2,1,1,1,1,1] => [8,4,4,1] => 43776
[10,1,1,1,1,1,1,1] => [9,4,4] => 2592
[9,8] => [4,4,3,3,3] => 2954880
[9,7,1] => [4,3,3,3,3,1] => 46539360
[9,6,2] => [6,3,3,3,1,1] => 23269680
[9,6,1,1] => [6,4,3,2,2] => 1969920
[9,5,3] => [5,4,2,2,2,1,1] => 505612800
[9,5,2,1] => [6,4,3,1,1,1,1] => 227525760
[9,5,1,1,1] => [6,4,2,2,2,1] => 27578880
[9,4,4] => [6,4,3,2,1,1] => 20684160
[9,4,1,1,1,1] => [7,4,3,2,1] => 1149120
[9,3,3,1,1] => [5,4,3,2,1,1,1] => 379209600
[9,3,2,2,1] => [6,3,3,2,2,1] => 31026240
[9,3,2,1,1,1] => [7,4,3,1,1,1] => 12065760
[9,3,1,1,1,1,1] => [8,4,3,1,1] => 656640
[9,2,2,2,2] => [5,4,3,3,2] => 2462400
[9,2,2,2,1,1] => [7,5,3,2] => 71820
[9,2,2,1,1,1,1] => [8,4,3,2] => 65664
[9,2,1,1,1,1,1,1] => [4,4,4,4,1] => 1751040
[9,1,1,1,1,1,1,1,1] => [5,4,4,4] => 109440
[8,8,1] => [6,4,3,3,1] => 1477440
[8,7,2] => [3,3,3,3,3,1,1] => 767899440
[8,7,1,1] => [4,3,3,3,2,2] => 62052480
[8,6,3] => [4,3,3,2,2,2,1] => 910103040
[8,6,1,1,1] => [5,4,2,2,2,2] => 45964800
[8,5,4] => [6,2,2,2,2,2,1] => 606735360
[8,5,2,1,1] => [6,3,2,2,2,1,1] => 455051520
[8,5,1,1,1,1] => [7,2,2,2,2,2] => 32175360
[8,4,2,1,1,1] => [7,4,2,1,1,1,1] => 176964480
[8,4,1,1,1,1,1] => [4,4,3,3,2,1] => 41368320
[8,3,3,3] => [5,3,3,2,2,1,1] => 568814400
[8,3,3,1,1,1] => [7,3,3,2,1,1] => 18098640
[8,3,2,2,2] => [5,4,3,3,1,1] => 25855200
[8,3,2,2,1,1] => [7,5,3,1,1] => 718200
[8,3,2,1,1,1,1] => [4,4,3,3,1,1,1] => 455051520
[8,3,1,1,1,1,1,1] => [4,4,4,3,1,1] => 27578880
[8,2,2,2,2,1] => [7,3,3,2,2] => 1723680
[8,2,2,2,1,1,1] => [5,5,3,2,2] => 2052000
[8,2,2,1,1,1,1,1] => [4,4,4,3,2] => 2626560
[8,2,1,1,1,1,1,1,1] => [9,4,3,1] => 36936
[8,1,1,1,1,1,1,1,1,1] => [10,4,3] => 2160
[7,7,3] => [4,4,3,2,2,2] => 55157760
[7,7,2,1] => [6,3,3,2,1,1,1] => 341288640
[7,7,1,1,1] => [7,4,2,2,2] => 1532160
[7,6,2,1,1] => [5,3,3,3,1,1,1] => 426610800
[7,5,1,1,1,1,1] => [4,3,2,2,2,2,2] => 1213470720
[7,4,1,1,1,1,1,1] => [8,4,2,2,1] => 875520
[7,3,3,2,1,1] => [4,4,3,2,2,1,1] => 606735360
[7,3,3,1,1,1,1] => [4,3,3,3,2,1,1] => 682577280
[7,3,2,2,1,1,1] => [5,4,3,2,2,1] => 34473600
[7,3,2,1,1,1,1,1] => [8,4,2,1,1,1] => 9192960
[7,3,1,1,1,1,1,1,1] => [9,3,3,1,1] => 554040
[7,2,2,2,2,2] => [3,3,3,2,2,2,2] => 1365154560
[7,2,2,2,2,1,1] => [5,3,3,2,2,2] => 51710400
[7,2,2,2,1,1,1,1] => [8,5,2,2] => 54720
[7,2,2,1,1,1,1,1,1] => [9,4,2,2] => 49248
[7,2,1,1,1,1,1,1,1,1] => [10,3,3,1] => 30780
[7,1,1,1,1,1,1,1,1,1,1] => [11,3,3] => 1782
[6,6,5] => [5,2,2,2,2,2,2] => 1011225600
[6,6,1,1,1,1,1] => [6,3,2,2,2,2] => 41368320
[6,5,2,1,1,1,1] => [7,3,3,1,1,1,1] => 199085040
[6,5,1,1,1,1,1,1] => [8,2,2,2,2,1] => 18385920
[6,4,2,1,1,1,1,1] => [8,3,1,1,1,1,1,1] => 1744364160
[6,4,1,1,1,1,1,1,1] => [9,2,2,2,1,1] => 10342080
[6,3,3,1,1,1,1,1] => [8,3,2,1,1,1,1] => 151683840
[6,3,2,2,2,2] => [5,3,2,2,2,2,1] => 758419200
[6,3,2,2,2,1,1] => [7,3,2,2,2,1] => 24131520
[6,3,2,2,1,1,1,1] => [8,3,3,1,1,1] => 10342080
[6,3,2,1,1,1,1,1,1] => [9,4,1,1,1,1] => 5171040
[6,3,1,1,1,1,1,1,1,1] => [10,3,2,1,1] => 410400
[6,2,2,2,2,2,1] => [5,4,4,2,2] => 2188800
[6,2,2,2,2,1,1,1] => [8,3,2,2,2] => 1313280
[6,2,2,2,1,1,1,1,1] => [9,2,2,2,2] => 984960
[6,2,2,1,1,1,1,1,1,1] => [10,4,2,1] => 27360
[6,2,1,1,1,1,1,1,1,1,1] => [11,3,2,1] => 22572
[6,1,1,1,1,1,1,1,1,1,1,1] => [12,3,2] => 1296
[5,5,2,1,1,1,1,1] => [8,4,1,1,1,1,1] => 101122560
[5,5,1,1,1,1,1,1,1] => [9,4,2,1,1] => 492480
[5,4,4,1,1,1,1] => [7,4,1,1,1,1,1,1] => 2035091520
[5,4,2,2,2,1,1] => [7,5,1,1,1,1,1] => 110602800
[5,4,1,1,1,1,1,1,1,1] => [10,2,2,1,1,1] => 5745600
[5,3,3,3,1,1,1] => [7,3,2,2,1,1,1] => 265446720
[5,3,3,2,2,2] => [4,4,4,2,1,1,1] => 404490240
[5,3,3,2,2,1,1] => [7,5,2,1,1,1] => 10054800
[5,3,3,1,1,1,1,1,1] => [9,2,1,1,1,1,1,1] => 1308273120
[5,3,2,2,2,2,1] => [5,4,4,1,1,1,1] => 252806400
[5,3,2,2,2,1,1,1] => [8,5,1,1,1,1] => 5745600
[5,3,2,2,1,1,1,1,1] => [9,3,1,1,1,1,1] => 85322160
[5,3,2,1,1,1,1,1,1,1] => [10,1,1,1,1,1,1,1] => 726818400
[5,3,1,1,1,1,1,1,1,1,1] => [11,2,1,1,1,1] => 3160080
[5,2,2,2,2,2,2] => [5,5,4,2,1] => 1368000
[5,2,2,2,2,2,1,1] => [8,6,2,1] => 32832
[5,2,2,2,2,1,1,1,1] => [9,5,2,1] => 30780
[5,2,2,2,1,1,1,1,1,1] => [10,2,2,2,1] => 547200
[5,2,2,1,1,1,1,1,1,1,1] => [11,2,2,1,1] => 300960
[5,2,1,1,1,1,1,1,1,1,1,1] => [12,3,1,1] => 12312
[5,1,1,1,1,1,1,1,1,1,1,1,1] => [13,2,2] => 936
[4,4,4,2,1,1,1] => [7,2,2,2,2,1,1] => 353928960
[4,4,4,1,1,1,1,1] => [8,3,2,2,1,1] => 13789440
[4,4,3,3,3] => [6,4,2,2,1,1,1] => 303367680
[4,4,3,3,2,1] => [4,4,2,2,2,2,1] => 808980480
[4,4,3,3,1,1,1] => [7,4,2,2,1,1] => 16087680
[4,4,3,2,2,2] => [4,4,4,2,2,1] => 36771840
[4,4,3,2,2,1,1] => [7,5,2,2,1] => 957600
[4,4,3,2,1,1,1,1] => [8,2,2,2,1,1,1] => 202245120
[4,4,3,1,1,1,1,1,1] => [9,2,2,1,1,1,1] => 113762880
[4,4,2,2,2,2,1] => [5,4,4,2,1,1] => 22982400
[4,4,2,2,2,1,1,1] => [8,5,2,1,1] => 547200
[4,4,2,2,1,1,1,1,1] => [9,3,2,1,1,1] => 7756560
[4,4,2,1,1,1,1,1,1,1] => [10,2,1,1,1,1,1] => 63201600
[4,4,1,1,1,1,1,1,1,1,1] => [11,3,1,1,1] => 225720
[4,3,3,3,3,1] => [3,3,3,3,2,2,1] => 1023865920
[4,3,3,3,2,2] => [5,3,3,3,2,1] => 38782800
[4,3,3,3,2,1,1] => [7,3,3,3,1] => 1292760
[4,3,3,3,1,1,1,1] => [8,3,3,2,1] => 984960
[4,3,3,2,2,2,1] => [5,4,4,3,1] => 1641600
[4,3,3,2,2,1,1,1] => [8,5,3,1] => 41040
[4,3,3,2,1,1,1,1,1] => [9,3,2,2,1] => 738720
[4,3,3,1,1,1,1,1,1,1] => [10,3,1,1,1,1] => 4309200
[4,3,2,2,2,2,2] => [5,5,4,1,1,1] => 14364000
[4,3,2,2,2,2,1,1] => [8,6,1,1,1] => 328320
[4,3,2,2,2,1,1,1,1] => [9,5,1,1,1] => 307800
[4,3,2,2,1,1,1,1,1,1] => [10,4,1,1,1] => 273600
[4,3,2,1,1,1,1,1,1,1,1] => [11,1,1,1,1,1,1] => 34760880
[4,3,1,1,1,1,1,1,1,1,1,1] => [12,1,1,1,1,1] => 1723680
[4,2,2,2,2,2,2,1] => [5,5,5,1,1] => 855000
[4,2,2,2,2,2,1,1,1] => [9,6,1,1] => 18468
[4,2,2,2,2,1,1,1,1,1] => [10,5,1,1] => 17100
[4,2,2,2,1,1,1,1,1,1,1] => [11,4,1,1] => 15048
[4,2,2,1,1,1,1,1,1,1,1,1] => [12,2,1,1,1] => 164160
[4,2,1,1,1,1,1,1,1,1,1,1,1] => [13,1,1,1,1] => 88920
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => [14,2,1] => 504
[3,3,3,3,3,2] => [3,3,3,3,3,2] => 69809040
[3,3,3,3,3,1,1] => [5,3,3,3,3] => 2770200
[3,3,3,3,2,2,1] => [6,4,4,3] => 98496
[3,3,3,3,2,1,1,1] => [8,3,3,3] => 73872
[3,3,3,3,1,1,1,1,1] => [9,3,3,2] => 55404
[3,3,3,2,2,2,2] => [5,5,4,3] => 102600
[3,3,3,2,2,2,1,1] => [8,6,3] => 2592
[3,3,3,2,2,1,1,1,1] => [9,5,3] => 2430
[3,3,3,2,1,1,1,1,1,1] => [10,3,2,2] => 41040
[3,3,3,1,1,1,1,1,1,1,1] => [11,2,2,2] => 30096
[3,3,2,2,2,2,2,1] => [5,5,5,2] => 85500
[3,3,2,2,2,2,1,1,1] => [9,6,2] => 1944
[3,3,2,2,2,1,1,1,1,1] => [10,5,2] => 1800
[3,3,2,2,1,1,1,1,1,1,1] => [11,4,2] => 1584
[3,3,2,1,1,1,1,1,1,1,1,1] => [12,2,2,1] => 16416
[3,3,1,1,1,1,1,1,1,1,1,1,1] => [13,2,1,1] => 8892
[3,2,2,2,2,2,2,2] => [6,5,5,1] => 51300
[3,2,2,2,2,2,2,1,1] => [9,7,1] => 1134
[3,2,2,2,2,2,1,1,1,1] => [10,6,1] => 1080
[3,2,2,2,2,1,1,1,1,1,1] => [11,5,1] => 990
[3,2,2,2,1,1,1,1,1,1,1,1] => [12,4,1] => 864
[3,2,2,1,1,1,1,1,1,1,1,1,1] => [13,3,1] => 702
[3,2,1,1,1,1,1,1,1,1,1,1,1,1] => [14,1,1,1] => 4788
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,1,1] => 270
[2,2,2,2,2,2,2,2,1] => [6,6,5] => 3240
[2,2,2,2,2,2,2,1,1,1] => [10,7] => 70
[2,2,2,2,2,2,1,1,1,1,1] => [11,6] => 66
[2,2,2,2,2,1,1,1,1,1,1,1] => [12,5] => 60
[2,2,2,2,1,1,1,1,1,1,1,1,1] => [13,4] => 52
[2,2,2,1,1,1,1,1,1,1,1,1,1,1] => [14,3] => 42
[2,2,1,1,1,1,1,1,1,1,1,1,1,1,1] => [15,2] => 30
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [16,1] => 16
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [17] => 1
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1,0,1 1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
$F_{1} = q$
$F_{2} = 2\ q$
$F_{3} = q + q^{2} + q^{4}$
$F_{4} = q + q^{3} + q^{4} + q^{10} + q^{30}$
$F_{5} = q + q^{4} + q^{6} + q^{18} + q^{24} + q^{84} + q^{336}$
$F_{6} = q + q^{5} + q^{8} + q^{9} + q^{28} + q^{42} + q^{56} + q^{168} + q^{224} + q^{1008} + q^{5040}$
$F_{7} = q + q^{6} + q^{10} + q^{12} + q^{40} + q^{64} + q^{72} + q^{96} + q^{288} + q^{432} + q^{576} + q^{2160} + q^{2880} + q^{15840} + q^{95040}$
$F_{8} = q + q^{7} + q^{12} + q^{15} + q^{16} + q^{54} + q^{90} + q^{108} + q^{144} + q^{162} + q^{450} + q^{720} + q^{810} + q^{1080} + q^{1440} + q^{3960} + q^{5940} + q^{7920} + q^{35640} + q^{47520} + q^{308880} + q^{2162160}$
$F_{9} = q + q^{8} + q^{14} + q^{18} + q^{20} + q^{70} + q^{120} + q^{150} + q^{160} + q^{200} + q^{240} + q^{270} + q^{660} + q^{1100} + q^{1320} + q^{1760} + q^{1980} + q^{2640} + q^{6600} + q^{10560} + q^{11880} + q^{15840} + q^{21120} + q^{68640} + q^{102960} + q^{137280} + q^{720720} + q^{960960} + q^{7207200} + q^{57657600}$
$F_{10} = q + q^{9} + q^{16} + q^{21} + q^{24} + q^{25} + q^{88} + q^{154} + q^{198} + q^{220} + q^{264} + q^{330} + q^{352} + q^{396} + q^{924} + q^{1584} + q^{1980} + q^{2112} + q^{2640} + q^{3168} + q^{3564} + q^{4224} + q^{4752} + q^{10296} + q^{17160} + q^{20592} + q^{27456} + q^{30888} + q^{41184} + q^{54912} + q^{120120} + q^{192192} + q^{216216} + q^{288288} + q^{384384} + q^{1441440} + q^{2162160} + q^{2882880} + q^{17297280} + q^{23063040} + q^{196035840} + q^{1764322560}$
Description
The number of transitive factorisations of a permutation of given cycle type into star transpositions.
Let $\pi$ be a permutation of cycle type $\lambda\vdash n$ and let $r=n + \ell(\lambda) - 2$. A minimal factorization of $\pi$ into star transpositions is an $r$-tuple of transpositions $(1, a_1)\dots(1, a_r)$ whose product (in this order) equals $\pi$.
The number of such factorizations equals [1]
$$ \frac{r!}{n!} \lambda_1\dots\lambda_{\ell(\lambda)}. $$
Let $\pi$ be a permutation of cycle type $\lambda\vdash n$ and let $r=n + \ell(\lambda) - 2$. A minimal factorization of $\pi$ into star transpositions is an $r$-tuple of transpositions $(1, a_1)\dots(1, a_r)$ whose product (in this order) equals $\pi$.
The number of such factorizations equals [1]
$$ \frac{r!}{n!} \lambda_1\dots\lambda_{\ell(\lambda)}. $$
Map
Loehr-Warrington inverse
Description
Return a partition whose length is the diagonal inversion number of the preimage.
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