Identifier
Values
=>
Cc0002;cc-rep
[]=>1
[1]=>1
[2]=>1
[1,1]=>2
[3]=>1
[2,1]=>3
[1,1,1]=>6
[4]=>1
[3,1]=>4
[2,2]=>6
[2,1,1]=>12
[1,1,1,1]=>24
[5]=>1
[4,1]=>5
[3,2]=>10
[3,1,1]=>20
[2,2,1]=>30
[2,1,1,1]=>60
[1,1,1,1,1]=>120
[6]=>1
[5,1]=>6
[4,2]=>15
[4,1,1]=>30
[3,3]=>20
[3,2,1]=>60
[3,1,1,1]=>120
[2,2,2]=>90
[2,2,1,1]=>180
[2,1,1,1,1]=>360
[1,1,1,1,1,1]=>720
[7]=>1
[6,1]=>7
[5,2]=>21
[5,1,1]=>42
[4,3]=>35
[4,2,1]=>105
[4,1,1,1]=>210
[3,3,1]=>140
[3,2,2]=>210
[3,2,1,1]=>420
[3,1,1,1,1]=>840
[2,2,2,1]=>630
[2,2,1,1,1]=>1260
[2,1,1,1,1,1]=>2520
[1,1,1,1,1,1,1]=>5040
[8]=>1
[7,1]=>8
[6,2]=>28
[6,1,1]=>56
[5,3]=>56
[5,2,1]=>168
[5,1,1,1]=>336
[4,4]=>70
[4,3,1]=>280
[4,2,2]=>420
[4,2,1,1]=>840
[4,1,1,1,1]=>1680
[3,3,2]=>560
[3,3,1,1]=>1120
[3,2,2,1]=>1680
[3,2,1,1,1]=>3360
[3,1,1,1,1,1]=>6720
[2,2,2,2]=>2520
[2,2,2,1,1]=>5040
[2,2,1,1,1,1]=>10080
[2,1,1,1,1,1,1]=>20160
[1,1,1,1,1,1,1,1]=>40320
[9]=>1
[8,1]=>9
[7,2]=>36
[7,1,1]=>72
[6,3]=>84
[6,2,1]=>252
[6,1,1,1]=>504
[5,4]=>126
[5,3,1]=>504
[5,2,2]=>756
[5,2,1,1]=>1512
[5,1,1,1,1]=>3024
[4,4,1]=>630
[4,3,2]=>1260
[4,3,1,1]=>2520
[4,2,2,1]=>3780
[4,2,1,1,1]=>7560
[4,1,1,1,1,1]=>15120
[3,3,3]=>1680
[3,3,2,1]=>5040
[3,3,1,1,1]=>10080
[3,2,2,2]=>7560
[3,2,2,1,1]=>15120
[3,2,1,1,1,1]=>30240
[3,1,1,1,1,1,1]=>60480
[2,2,2,2,1]=>22680
[2,2,2,1,1,1]=>45360
[2,2,1,1,1,1,1]=>90720
[2,1,1,1,1,1,1,1]=>181440
[1,1,1,1,1,1,1,1,1]=>362880
[10]=>1
[9,1]=>10
[8,2]=>45
[8,1,1]=>90
[7,3]=>120
[7,2,1]=>360
[7,1,1,1]=>720
[6,4]=>210
[6,3,1]=>840
[6,2,2]=>1260
[6,2,1,1]=>2520
[6,1,1,1,1]=>5040
[5,5]=>252
[5,4,1]=>1260
[5,3,2]=>2520
[5,3,1,1]=>5040
[5,2,2,1]=>7560
[5,2,1,1,1]=>15120
[5,1,1,1,1,1]=>30240
[4,4,2]=>3150
[4,4,1,1]=>6300
[4,3,3]=>4200
[4,3,2,1]=>12600
[4,3,1,1,1]=>25200
[4,2,2,2]=>18900
[4,2,2,1,1]=>37800
[4,2,1,1,1,1]=>75600
[4,1,1,1,1,1,1]=>151200
[3,3,3,1]=>16800
[3,3,2,2]=>25200
[3,3,2,1,1]=>50400
[3,3,1,1,1,1]=>100800
[3,2,2,2,1]=>75600
[3,2,2,1,1,1]=>151200
[3,2,1,1,1,1,1]=>302400
[3,1,1,1,1,1,1,1]=>604800
[2,2,2,2,2]=>113400
[2,2,2,2,1,1]=>226800
[2,2,2,1,1,1,1]=>453600
[2,2,1,1,1,1,1,1]=>907200
[2,1,1,1,1,1,1,1,1]=>1814400
[1,1,1,1,1,1,1,1,1,1]=>3628800
[11]=>1
[10,1]=>11
[9,2]=>55
[9,1,1]=>110
[8,3]=>165
[8,2,1]=>495
[8,1,1,1]=>990
[7,4]=>330
[7,3,1]=>1320
[7,2,2]=>1980
[7,2,1,1]=>3960
[7,1,1,1,1]=>7920
[6,5]=>462
[6,4,1]=>2310
[6,3,2]=>4620
[6,3,1,1]=>9240
[6,2,2,1]=>13860
[6,2,1,1,1]=>27720
[6,1,1,1,1,1]=>55440
[5,5,1]=>2772
[5,4,2]=>6930
[5,4,1,1]=>13860
[5,3,3]=>9240
[5,3,2,1]=>27720
[5,3,1,1,1]=>55440
[5,2,2,2]=>41580
[5,2,2,1,1]=>83160
[5,2,1,1,1,1]=>166320
[5,1,1,1,1,1,1]=>332640
[4,4,3]=>11550
[4,4,2,1]=>34650
[4,4,1,1,1]=>69300
[4,3,3,1]=>46200
[4,3,2,2]=>69300
[4,3,2,1,1]=>138600
[4,3,1,1,1,1]=>277200
[4,2,2,2,1]=>207900
[4,2,2,1,1,1]=>415800
[4,2,1,1,1,1,1]=>831600
[4,1,1,1,1,1,1,1]=>1663200
[3,3,3,2]=>92400
[3,3,3,1,1]=>184800
[3,3,2,2,1]=>277200
[3,3,2,1,1,1]=>554400
[3,3,1,1,1,1,1]=>1108800
[3,2,2,2,2]=>415800
[3,2,2,2,1,1]=>831600
[3,2,2,1,1,1,1]=>1663200
[3,2,1,1,1,1,1,1]=>3326400
[3,1,1,1,1,1,1,1,1]=>6652800
[2,2,2,2,2,1]=>1247400
[2,2,2,2,1,1,1]=>2494800
[2,2,2,1,1,1,1,1]=>4989600
[2,2,1,1,1,1,1,1,1]=>9979200
[12]=>1
[11,1]=>12
[10,2]=>66
[10,1,1]=>132
[9,3]=>220
[9,2,1]=>660
[9,1,1,1]=>1320
[8,4]=>495
[8,3,1]=>1980
[8,2,2]=>2970
[8,2,1,1]=>5940
[8,1,1,1,1]=>11880
[7,5]=>792
[7,4,1]=>3960
[7,3,2]=>7920
[7,3,1,1]=>15840
[7,2,2,1]=>23760
[7,2,1,1,1]=>47520
[7,1,1,1,1,1]=>95040
[6,6]=>924
[6,5,1]=>5544
[6,4,2]=>13860
[6,4,1,1]=>27720
[6,3,3]=>18480
[6,3,2,1]=>55440
[6,3,1,1,1]=>110880
[6,2,2,2]=>83160
[6,2,2,1,1]=>166320
[6,2,1,1,1,1]=>332640
[6,1,1,1,1,1,1]=>665280
[5,5,2]=>16632
[5,5,1,1]=>33264
[5,4,3]=>27720
[5,4,2,1]=>83160
[5,4,1,1,1]=>166320
[5,3,3,1]=>110880
[5,3,2,2]=>166320
[5,3,2,1,1]=>332640
[5,3,1,1,1,1]=>665280
[5,2,2,2,1]=>498960
[5,2,2,1,1,1]=>997920
[5,2,1,1,1,1,1]=>1995840
[5,1,1,1,1,1,1,1]=>3991680
[4,4,4]=>34650
[4,4,3,1]=>138600
[4,4,2,2]=>207900
[4,4,2,1,1]=>415800
[4,4,1,1,1,1]=>831600
[4,3,3,2]=>277200
[4,3,3,1,1]=>554400
[4,3,2,2,1]=>831600
[4,3,2,1,1,1]=>1663200
[4,3,1,1,1,1,1]=>3326400
[4,2,2,2,2]=>1247400
[4,2,2,2,1,1]=>2494800
[4,2,2,1,1,1,1]=>4989600
[4,2,1,1,1,1,1,1]=>9979200
[3,3,3,3]=>369600
[3,3,3,2,1]=>1108800
[3,3,3,1,1,1]=>2217600
[3,3,2,2,2]=>1663200
[3,3,2,2,1,1]=>3326400
[3,3,2,1,1,1,1]=>6652800
[3,2,2,2,2,1]=>4989600
[3,2,2,2,1,1,1]=>9979200
[2,2,2,2,2,2]=>7484400
[13]=>1
[12,1]=>13
[11,2]=>78
[11,1,1]=>156
[10,3]=>286
[10,2,1]=>858
[10,1,1,1]=>1716
[9,4]=>715
[9,3,1]=>2860
[9,2,2]=>4290
[9,2,1,1]=>8580
[9,1,1,1,1]=>17160
[8,5]=>1287
[8,4,1]=>6435
[8,3,2]=>12870
[8,3,1,1]=>25740
[8,2,2,1]=>38610
[8,2,1,1,1]=>77220
[8,1,1,1,1,1]=>154440
[7,6]=>1716
[7,5,1]=>10296
[7,4,2]=>25740
[7,4,1,1]=>51480
[7,3,3]=>34320
[7,3,2,1]=>102960
[7,3,1,1,1]=>205920
[7,2,2,2]=>154440
[7,2,2,1,1]=>308880
[7,2,1,1,1,1]=>617760
[7,1,1,1,1,1,1]=>1235520
[6,6,1]=>12012
[6,5,2]=>36036
[6,5,1,1]=>72072
[6,4,3]=>60060
[6,4,2,1]=>180180
[6,4,1,1,1]=>360360
[6,3,3,1]=>240240
[6,3,2,2]=>360360
[6,3,2,1,1]=>720720
[6,3,1,1,1,1]=>1441440
[6,2,2,2,1]=>1081080
[6,2,2,1,1,1]=>2162160
[6,2,1,1,1,1,1]=>4324320
[6,1,1,1,1,1,1,1]=>8648640
[5,5,3]=>72072
[5,5,2,1]=>216216
[5,5,1,1,1]=>432432
[5,4,4]=>90090
[5,4,3,1]=>360360
[5,4,2,2]=>540540
[5,4,2,1,1]=>1081080
[5,4,1,1,1,1]=>2162160
[5,3,3,2]=>720720
[5,3,3,1,1]=>1441440
[5,3,2,2,1]=>2162160
[5,3,2,1,1,1]=>4324320
[5,3,1,1,1,1,1]=>8648640
[5,2,2,2,2]=>3243240
[5,2,2,2,1,1]=>6486480
[4,4,4,1]=>450450
[4,4,3,2]=>900900
[4,4,3,1,1]=>1801800
[4,4,2,2,1]=>2702700
[4,4,2,1,1,1]=>5405400
[4,3,3,3]=>1201200
[4,3,3,2,1]=>3603600
[4,3,3,1,1,1]=>7207200
[4,3,2,2,2]=>5405400
[3,3,3,3,1]=>4804800
[3,3,3,2,2]=>7207200
[14]=>1
[13,1]=>14
[12,2]=>91
[12,1,1]=>182
[11,3]=>364
[11,2,1]=>1092
[11,1,1,1]=>2184
[10,4]=>1001
[10,3,1]=>4004
[10,2,2]=>6006
[10,2,1,1]=>12012
[10,1,1,1,1]=>24024
[9,5]=>2002
[9,4,1]=>10010
[9,3,2]=>20020
[9,3,1,1]=>40040
[9,2,2,1]=>60060
[9,2,1,1,1]=>120120
[9,1,1,1,1,1]=>240240
[8,6]=>3003
[8,5,1]=>18018
[8,4,2]=>45045
[8,4,1,1]=>90090
[8,3,3]=>60060
[8,3,2,1]=>180180
[8,3,1,1,1]=>360360
[8,2,2,2]=>270270
[8,2,2,1,1]=>540540
[8,2,1,1,1,1]=>1081080
[8,1,1,1,1,1,1]=>2162160
[7,7]=>3432
[7,6,1]=>24024
[7,5,2]=>72072
[7,5,1,1]=>144144
[7,4,3]=>120120
[7,4,2,1]=>360360
[7,4,1,1,1]=>720720
[7,3,3,1]=>480480
[7,3,2,2]=>720720
[7,3,2,1,1]=>1441440
[7,3,1,1,1,1]=>2882880
[7,2,2,2,1]=>2162160
[7,2,2,1,1,1]=>4324320
[7,2,1,1,1,1,1]=>8648640
[6,6,2]=>84084
[6,6,1,1]=>168168
[6,5,3]=>168168
[6,5,2,1]=>504504
[6,5,1,1,1]=>1009008
[6,4,4]=>210210
[6,4,3,1]=>840840
[6,4,2,2]=>1261260
[6,4,2,1,1]=>2522520
[6,4,1,1,1,1]=>5045040
[6,3,3,2]=>1681680
[6,3,3,1,1]=>3363360
[6,3,2,2,1]=>5045040
[6,2,2,2,2]=>7567560
[5,5,4]=>252252
[5,5,3,1]=>1009008
[5,5,2,2]=>1513512
[5,5,2,1,1]=>3027024
[5,5,1,1,1,1]=>6054048
[5,4,4,1]=>1261260
[5,4,3,2]=>2522520
[5,4,3,1,1]=>5045040
[5,4,2,2,1]=>7567560
[5,3,3,3]=>3363360
[4,4,4,2]=>3153150
[4,4,4,1,1]=>6306300
[4,4,3,3]=>4204200
[15]=>1
[14,1]=>15
[13,2]=>105
[13,1,1]=>210
[12,3]=>455
[12,2,1]=>1365
[12,1,1,1]=>2730
[11,4]=>1365
[11,3,1]=>5460
[11,2,2]=>8190
[11,2,1,1]=>16380
[11,1,1,1,1]=>32760
[10,5]=>3003
[10,4,1]=>15015
[10,3,2]=>30030
[10,3,1,1]=>60060
[10,2,2,1]=>90090
[10,2,1,1,1]=>180180
[10,1,1,1,1,1]=>360360
[9,6]=>5005
[9,5,1]=>30030
[9,4,2]=>75075
[9,4,1,1]=>150150
[9,3,3]=>100100
[9,3,2,1]=>300300
[9,3,1,1,1]=>600600
[9,2,2,2]=>450450
[9,2,2,1,1]=>900900
[9,2,1,1,1,1]=>1801800
[9,1,1,1,1,1,1]=>3603600
[8,7]=>6435
[8,6,1]=>45045
[8,5,2]=>135135
[8,5,1,1]=>270270
[8,4,3]=>225225
[8,4,2,1]=>675675
[8,4,1,1,1]=>1351350
[8,3,3,1]=>900900
[8,3,2,2]=>1351350
[8,3,2,1,1]=>2702700
[8,3,1,1,1,1]=>5405400
[8,2,2,2,1]=>4054050
[8,2,2,1,1,1]=>8108100
[7,7,1]=>51480
[7,6,2]=>180180
[7,6,1,1]=>360360
[7,5,3]=>360360
[7,5,2,1]=>1081080
[7,5,1,1,1]=>2162160
[7,4,4]=>450450
[7,4,3,1]=>1801800
[7,4,2,2]=>2702700
[7,4,2,1,1]=>5405400
[7,3,3,2]=>3603600
[7,3,3,1,1]=>7207200
[6,6,3]=>420420
[6,6,2,1]=>1261260
[6,6,1,1,1]=>2522520
[6,5,4]=>630630
[6,5,3,1]=>2522520
[6,5,2,2]=>3783780
[6,5,2,1,1]=>7567560
[6,4,4,1]=>3153150
[6,4,3,2]=>6306300
[6,3,3,3]=>8408400
[5,5,5]=>756756
[5,5,4,1]=>3783780
[5,5,3,2]=>7567560
[5,4,4,2]=>9459450
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Description
The multinomial of the parts of a partition.
Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial
$$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$
For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$.
Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial
$$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$
For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$.
Code
def statistic(L):
return multinomial(list(L))
Created
Mar 25, 2013 at 10:25 by Christian Stump
Updated
Oct 29, 2017 at 20:26 by Martin Rubey
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