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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St000517
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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 $\frac{1}{n+1}\binom{2n}{n}$.
-----------------------------------------------------------------------------
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)
-----------------------------------------------------------------------------
Statistic values:
[] => 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
-----------------------------------------------------------------------------
Created: May 31, 2016 at 14:57 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Jun 19, 2023 at 10:40 by Martin Rubey