Identifier
Values
=>
Cc0002;cc-rep
[]=>1
[1]=>1
[2]=>1
[1,1]=>1
[3]=>1
[2,1]=>3
[1,1,1]=>1
[4]=>1
[3,1]=>4
[2,2]=>2
[2,1,1]=>6
[1,1,1,1]=>1
[5]=>1
[4,1]=>5
[3,2]=>5
[3,1,1]=>10
[2,2,1]=>10
[2,1,1,1]=>10
[1,1,1,1,1]=>1
[6]=>1
[5,1]=>6
[4,2]=>6
[4,1,1]=>15
[3,3]=>3
[3,2,1]=>30
[3,1,1,1]=>20
[2,2,2]=>5
[2,2,1,1]=>30
[2,1,1,1,1]=>15
[1,1,1,1,1,1]=>1
[7]=>1
[6,1]=>7
[5,2]=>7
[5,1,1]=>21
[4,3]=>7
[4,2,1]=>42
[4,1,1,1]=>35
[3,3,1]=>21
[3,2,2]=>21
[3,2,1,1]=>105
[3,1,1,1,1]=>35
[2,2,2,1]=>35
[2,2,1,1,1]=>70
[2,1,1,1,1,1]=>21
[1,1,1,1,1,1,1]=>1
[8]=>1
[7,1]=>8
[6,2]=>8
[6,1,1]=>28
[5,3]=>8
[5,2,1]=>56
[5,1,1,1]=>56
[4,4]=>4
[4,3,1]=>56
[4,2,2]=>28
[4,2,1,1]=>168
[4,1,1,1,1]=>70
[3,3,2]=>28
[3,3,1,1]=>84
[3,2,2,1]=>168
[3,2,1,1,1]=>280
[3,1,1,1,1,1]=>56
[2,2,2,2]=>14
[2,2,2,1,1]=>140
[2,2,1,1,1,1]=>140
[2,1,1,1,1,1,1]=>28
[1,1,1,1,1,1,1,1]=>1
[9]=>1
[8,1]=>9
[7,2]=>9
[7,1,1]=>36
[6,3]=>9
[6,2,1]=>72
[6,1,1,1]=>84
[5,4]=>9
[5,3,1]=>72
[5,2,2]=>36
[5,2,1,1]=>252
[5,1,1,1,1]=>126
[4,4,1]=>36
[4,3,2]=>72
[4,3,1,1]=>252
[4,2,2,1]=>252
[4,2,1,1,1]=>504
[4,1,1,1,1,1]=>126
[3,3,3]=>12
[3,3,2,1]=>252
[3,3,1,1,1]=>252
[3,2,2,2]=>84
[3,2,2,1,1]=>756
[3,2,1,1,1,1]=>630
[3,1,1,1,1,1,1]=>84
[2,2,2,2,1]=>126
[2,2,2,1,1,1]=>420
[2,2,1,1,1,1,1]=>252
[2,1,1,1,1,1,1,1]=>36
[1,1,1,1,1,1,1,1,1]=>1
[10]=>1
[9,1]=>10
[8,2]=>10
[8,1,1]=>45
[7,3]=>10
[7,2,1]=>90
[7,1,1,1]=>120
[6,4]=>10
[6,3,1]=>90
[6,2,2]=>45
[6,2,1,1]=>360
[6,1,1,1,1]=>210
[5,5]=>5
[5,4,1]=>90
[5,3,2]=>90
[5,3,1,1]=>360
[5,2,2,1]=>360
[5,2,1,1,1]=>840
[5,1,1,1,1,1]=>252
[4,4,2]=>45
[4,4,1,1]=>180
[4,3,3]=>45
[4,3,2,1]=>720
[4,3,1,1,1]=>840
[4,2,2,2]=>120
[4,2,2,1,1]=>1260
[4,2,1,1,1,1]=>1260
[4,1,1,1,1,1,1]=>210
[3,3,3,1]=>120
[3,3,2,2]=>180
[3,3,2,1,1]=>1260
[3,3,1,1,1,1]=>630
[3,2,2,2,1]=>840
[3,2,2,1,1,1]=>2520
[3,2,1,1,1,1,1]=>1260
[3,1,1,1,1,1,1,1]=>120
[2,2,2,2,2]=>42
[2,2,2,2,1,1]=>630
[2,2,2,1,1,1,1]=>1050
[2,2,1,1,1,1,1,1]=>420
[2,1,1,1,1,1,1,1,1]=>45
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>1
[10,1]=>11
[9,2]=>11
[9,1,1]=>55
[8,3]=>11
[8,2,1]=>110
[8,1,1,1]=>165
[7,4]=>11
[7,3,1]=>110
[7,2,2]=>55
[7,2,1,1]=>495
[7,1,1,1,1]=>330
[6,5]=>11
[6,4,1]=>110
[6,3,2]=>110
[6,3,1,1]=>495
[6,2,2,1]=>495
[6,2,1,1,1]=>1320
[6,1,1,1,1,1]=>462
[5,5,1]=>55
[5,4,2]=>110
[5,4,1,1]=>495
[5,3,3]=>55
[5,3,2,1]=>990
[5,3,1,1,1]=>1320
[5,2,2,2]=>165
[5,2,2,1,1]=>1980
[5,2,1,1,1,1]=>2310
[5,1,1,1,1,1,1]=>462
[4,4,3]=>55
[4,4,2,1]=>495
[4,4,1,1,1]=>660
[4,3,3,1]=>495
[4,3,2,2]=>495
[4,3,2,1,1]=>3960
[4,3,1,1,1,1]=>2310
[4,2,2,2,1]=>1320
[4,2,2,1,1,1]=>4620
[4,2,1,1,1,1,1]=>2772
[4,1,1,1,1,1,1,1]=>330
[3,3,3,2]=>165
[3,3,3,1,1]=>660
[3,3,2,2,1]=>1980
[3,3,2,1,1,1]=>4620
[3,3,1,1,1,1,1]=>1386
[3,2,2,2,2]=>330
[3,2,2,2,1,1]=>4620
[3,2,2,1,1,1,1]=>6930
[3,2,1,1,1,1,1,1]=>2310
[3,1,1,1,1,1,1,1,1]=>165
[2,2,2,2,2,1]=>462
[2,2,2,2,1,1,1]=>2310
[2,2,2,1,1,1,1,1]=>2310
[2,2,1,1,1,1,1,1,1]=>660
[2,1,1,1,1,1,1,1,1,1]=>55
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>1
[11,1]=>12
[10,2]=>12
[10,1,1]=>66
[9,3]=>12
[9,2,1]=>132
[9,1,1,1]=>220
[8,4]=>12
[8,3,1]=>132
[8,2,2]=>66
[8,2,1,1]=>660
[8,1,1,1,1]=>495
[7,5]=>12
[7,4,1]=>132
[7,3,2]=>132
[7,3,1,1]=>660
[7,2,2,1]=>660
[7,2,1,1,1]=>1980
[7,1,1,1,1,1]=>792
[6,6]=>6
[6,5,1]=>132
[6,4,2]=>132
[6,4,1,1]=>660
[6,3,3]=>66
[6,3,2,1]=>1320
[6,3,1,1,1]=>1980
[6,2,2,2]=>220
[6,2,2,1,1]=>2970
[6,2,1,1,1,1]=>3960
[6,1,1,1,1,1,1]=>924
[5,5,2]=>66
[5,5,1,1]=>330
[5,4,3]=>132
[5,4,2,1]=>1320
[5,4,1,1,1]=>1980
[5,3,3,1]=>660
[5,3,2,2]=>660
[5,3,2,1,1]=>5940
[5,3,1,1,1,1]=>3960
[5,2,2,2,1]=>1980
[5,2,2,1,1,1]=>7920
[5,2,1,1,1,1,1]=>5544
[5,1,1,1,1,1,1,1]=>792
[4,4,4]=>22
[4,4,3,1]=>660
[4,4,2,2]=>330
[4,4,2,1,1]=>2970
[4,4,1,1,1,1]=>1980
[4,3,3,2]=>660
[4,3,3,1,1]=>2970
[4,3,2,2,1]=>5940
[4,3,2,1,1,1]=>15840
[4,3,1,1,1,1,1]=>5544
[4,2,2,2,2]=>495
[4,2,2,2,1,1]=>7920
[4,2,2,1,1,1,1]=>13860
[4,2,1,1,1,1,1,1]=>5544
[4,1,1,1,1,1,1,1,1]=>495
[3,3,3,3]=>55
[3,3,3,2,1]=>1980
[3,3,3,1,1,1]=>2640
[3,3,2,2,2]=>990
[3,3,2,2,1,1]=>11880
[3,3,2,1,1,1,1]=>13860
[3,3,1,1,1,1,1,1]=>2772
[3,2,2,2,2,1]=>3960
[3,2,2,2,1,1,1]=>18480
[3,2,2,1,1,1,1,1]=>16632
[3,2,1,1,1,1,1,1,1]=>3960
[3,1,1,1,1,1,1,1,1,1]=>220
[2,2,2,2,2,2]=>132
[2,2,2,2,2,1,1]=>2772
[2,2,2,2,1,1,1,1]=>6930
[2,2,2,1,1,1,1,1,1]=>4620
[2,2,1,1,1,1,1,1,1,1]=>990
[2,1,1,1,1,1,1,1,1,1,1]=>66
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
[13]=>1
[12,1]=>13
[10,3]=>13
[8,5]=>13
[7,6]=>13
[7,5,1]=>156
[7,4,2]=>156
[6,6,1]=>78
[6,4,2,1]=>1716
[5,5,3]=>78
[5,4,4]=>78
[5,4,3,1]=>1716
[5,4,2,2]=>858
[5,4,2,1,1]=>8580
[5,4,1,1,1,1]=>6435
[5,3,3,2]=>858
[5,3,3,1,1]=>4290
[5,3,2,2,1]=>8580
[5,3,2,1,1,1]=>25740
[4,4,4,1]=>286
[4,4,3,2]=>858
[4,4,3,1,1]=>4290
[4,4,2,2,1]=>4290
[4,3,3,3]=>286
[4,3,3,2,1]=>8580
[3,3,3,3,1]=>715
[3,3,3,2,2]=>1430
[3,3,2,2,2,1]=>12870
[3,2,2,2,2,2]=>1287
[2,2,2,2,2,2,1]=>1716
[1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[14]=>1
[13,1]=>14
[9,5]=>14
[8,5,1]=>182
[7,7]=>7
[7,5,2]=>182
[7,4,3]=>182
[6,6,2]=>91
[6,4,4]=>91
[6,2,2,2,2]=>1001
[5,5,4]=>91
[5,5,1,1,1,1]=>5005
[5,4,3,2]=>2184
[5,4,3,1,1]=>12012
[5,4,2,2,1]=>12012
[5,4,2,1,1,1]=>40040
[5,3,3,3]=>364
[5,3,3,2,1]=>12012
[5,2,2,2,2,1]=>10010
[4,4,4,2]=>364
[4,4,3,3]=>546
[4,4,3,2,1]=>12012
[4,3,2,2,2,1]=>40040
[3,3,3,3,2]=>1001
[3,3,3,3,1,1]=>5005
[3,3,2,2,2,2]=>5005
[2,2,2,2,2,2,2]=>429
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[15]=>1
[14,1]=>15
[9,5,1]=>210
[8,5,2]=>210
[7,5,3]=>210
[6,6,3]=>105
[6,5,4]=>210
[6,5,1,1,1,1]=>15015
[6,3,3,3]=>455
[6,2,2,2,2,1]=>15015
[5,5,5]=>35
[5,4,3,3]=>1365
[5,4,3,2,1]=>32760
[5,4,3,1,1,1]=>60060
[5,3,2,2,2,1]=>60060
[4,4,4,3]=>455
[4,4,4,1,1,1]=>10010
[4,3,3,3,2]=>5460
[3,3,3,3,3]=>273
[3,3,3,3,2,1]=>15015
[3,3,3,2,2,2]=>10010
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[16]=>1
[15,1]=>16
[8,8]=>8
[8,5,3]=>240
[7,5,3,1]=>3360
[6,6,4]=>120
[5,5,3,3]=>840
[5,5,2,2,2]=>3640
[5,4,4,3]=>1680
[5,4,3,2,1,1]=>262080
[5,4,2,2,2,1]=>87360
[4,4,4,4]=>140
[4,4,4,2,2]=>3640
[4,4,3,3,2]=>10920
[4,3,3,3,3]=>1820
[4,3,3,3,2,1]=>87360
[3,3,3,3,2,2]=>10920
[2,2,2,2,2,2,2,2]=>1430
[17]=>1
[8,6,3]=>272
[7,5,3,2]=>4080
[6,5,3,3]=>2040
[6,5,2,2,2]=>9520
[6,4,4,3]=>2040
[6,4,4,1,1,1]=>61880
[6,3,3,3,2]=>9520
[6,3,3,3,1,1]=>61880
[5,5,4,3]=>2040
[5,5,4,1,1,1]=>61880
[5,5,2,2,2,1]=>61880
[5,4,4,4]=>680
[5,4,3,2,2,1]=>371280
[5,3,3,3,2,1]=>123760
[4,4,4,3,2]=>9520
[4,4,4,3,1,1]=>61880
[4,4,4,2,2,1]=>61880
[4,4,3,3,3]=>4760
[3,3,3,3,3,2]=>6188
[4,4,4,3,2,1]=>171360
[5,4,3,3,2,1]=>514080
[6,3,3,3,2,1]=>171360
[6,5,2,2,2,1]=>171360
[5,5,3,3,1,1]=>128520
[6,5,4,1,1,1]=>171360
[5,5,3,3,2]=>18360
[5,5,4,2,2]=>18360
[6,4,4,2,2]=>18360
[6,5,4,3]=>4896
[4,4,4,3,3]=>6120
[4,4,4,4,2]=>3060
[5,5,4,4]=>1224
[2,2,2,2,2,2,2,2,2]=>4862
[3,3,3,3,3,3]=>1428
[18]=>1
[6,6,6]=>51
[9,6,3]=>306
[8,6,4]=>306
[9,9]=>9
[5,4,4,3,2,1]=>697680
[5,5,3,3,2,1]=>348840
[5,5,4,2,2,1]=>348840
[6,4,4,2,2,1]=>348840
[5,5,4,3,1,1]=>348840
[6,4,4,3,1,1]=>348840
[6,5,3,3,1,1]=>348840
[5,5,4,3,2]=>46512
[6,4,4,3,2]=>46512
[6,5,3,3,2]=>46512
[6,5,4,2,2]=>46512
[6,5,4,3,1]=>93024
[6,5,4,1,1,1,1]=>813960
[4,4,4,4,3]=>3876
[4,3,3,3,3,3]=>11628
[19]=>1
[9,6,4]=>342
[8,5,4,2]=>5814
[8,5,5,1]=>2907
[5,5,4,3,2,1]=>930240
[6,4,4,3,2,1]=>930240
[6,5,3,3,2,1]=>930240
[6,5,4,2,2,1]=>930240
[6,5,4,3,1,1]=>930240
[6,5,4,3,2]=>116280
[6,5,2,2,2,2,1]=>1162800
[6,5,4,2,1,1,1]=>4651200
[7,5,4,3,1]=>116280
[3,3,3,3,3,3,2]=>38760
[4,4,3,3,3,3]=>38760
[4,4,4,4,4]=>969
[5,5,5,5]=>285
[20]=>1
[8,6,4,2]=>6840
[10,6,4]=>380
[10,7,3]=>380
[9,7,4]=>380
[9,5,5,1]=>3420
[6,5,4,3,2,1]=>2441880
[6,3,3,3,3,2,1]=>1627920
[6,5,3,2,2,2,1]=>6511680
[6,5,4,3,1,1,1]=>6511680
[3,3,3,3,3,3,3]=>7752
[4,4,4,3,3,3]=>67830
[21]=>1
[11,7,3]=>420
[4,4,4,4,3,2,1]=>2238390
[6,4,3,3,3,2,1]=>8953560
[6,5,4,2,2,2,1]=>8953560
[6,5,4,3,2,1,1]=>26860680
[4,4,4,4,3,3]=>65835
[9,6,4,3]=>9240
[5,4,4,4,3,2,1]=>12113640
[6,5,3,3,3,2,1]=>12113640
[6,5,4,3,2,2,1]=>36340920
[9,6,5,3]=>10626
[8,6,5,3,1]=>212520
[6,4,4,4,3,2,1]=>16151520
[6,5,4,3,3,2,1]=>48454560
[3,3,3,3,3,3,3,3]=>43263
[4,4,4,4,4,4]=>7084
[11,7,5,1]=>12144
[9,7,5,3]=>12144
[8,8,8]=>92
[5,5,5,4,3,2,1]=>21252000
[6,5,4,4,3,2,1]=>63756000
[9,7,5,3,1]=>303600
[10,7,5,3]=>13800
[6,5,5,4,3,2,1]=>82882800
[9,7,5,4,1]=>358800
[6,6,5,4,3,2,1]=>106563600
[7,6,5,4,3,2]=>9687600
[3,3,3,3,3,3,3,3,3]=>246675
[7,6,5,4,3,2,1]=>271252800
[7,6,5,4,3,1,1,1]=>994593600
[10,7,6,4,1]=>491400
[9,7,6,4,2]=>491400
[10,8,5,4,1]=>491400
[7,6,5,4,2,2,2,1]=>1311055200
[10,8,6,4,1]=>570024
[9,7,5,5,3,1]=>8550360
[7,6,5,3,3,3,2,1]=>1710072000
[11,8,6,4,1]=>657720
[10,8,6,4,2]=>657720
[11,8,6,5,1]=>755160
[4,4,4,4,4,4,4,4]=>420732
[12,9,7,5,1]=>1113024
[13,9,7,5,1]=>1256640
[11,9,7,5,3,1]=>45239040
[11,8,7,5,4,1]=>45239040
[11,9,7,5,5,3]=>39480480
[11,9,7,7,5,3,3]=>1466110800
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Description
The Kreweras number of an integer partition.
This is defined for $\lambda \vdash n$ with $k$ parts as
$$\frac{1}{n+1}\binom{n+1}{n+1-k,\mu_1(\lambda),\ldots,\mu_n(\lambda)}$$
where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$, see [1]. This formula indeed counts the number of noncrossing set partitions where the ordered block sizes are the partition $\lambda$.
These numbers refine the Narayana numbers $N(n,k) = \frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1}$ and thus sum up to the Catalan numbers $\frac{1}{n+1}\binom{2n}{n}$.
This is defined for $\lambda \vdash n$ with $k$ parts as
$$\frac{1}{n+1}\binom{n+1}{n+1-k,\mu_1(\lambda),\ldots,\mu_n(\lambda)}$$
where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$, see [1]. This formula indeed counts the number of noncrossing set partitions where the ordered block sizes are the partition $\lambda$.
These numbers refine the Narayana numbers $N(n,k) = \frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1}$ and thus sum up to the Catalan numbers $\frac{1}{n+1}\binom{2n}{n}$.
References
[1] Reiner, V., Sommers, E. Weyl group $q$-Kreweras numbers and cyclic sieving arXiv:1605.09172
Code
def statistic(la):
la = list(la)
n = sum(la)
k = len(la)
multi = [n+1-k]+[ la.count(j) for j in [1..n] ]
return multinomial(multi)/(n+1)
Created
May 31, 2016 at 14:57 by Christian Stump
Updated
Jun 19, 2023 at 10:40 by Martin Rubey
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