Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 1
[2,1] generating graphics... => 3
[1,1,1] generating graphics... => 1
[4] generating graphics... => 1
[3,1] generating graphics... => 4
[2,2] generating graphics... => 2
[2,1,1] generating graphics... => 6
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 1
[4,1] generating graphics... => 5
[3,2] generating graphics... => 5
[3,1,1] generating graphics... => 10
[2,2,1] generating graphics... => 10
[2,1,1,1] generating graphics... => 10
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 1
[5,1] generating graphics... => 6
[4,2] generating graphics... => 6
[4,1,1] generating graphics... => 15
[3,3] generating graphics... => 3
[3,2,1] generating graphics... => 30
[3,1,1,1] generating graphics... => 20
[2,2,2] generating graphics... => 5
[2,2,1,1] generating graphics... => 30
[2,1,1,1,1] generating graphics... => 15
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 1
[6,1] generating graphics... => 7
[5,2] generating graphics... => 7
[5,1,1] generating graphics... => 21
[4,3] generating graphics... => 7
[4,2,1] generating graphics... => 42
[4,1,1,1] generating graphics... => 35
[3,3,1] generating graphics... => 21
[3,2,2] generating graphics... => 21
[3,2,1,1] generating graphics... => 105
[3,1,1,1,1] generating graphics... => 35
[2,2,2,1] generating graphics... => 35
[2,2,1,1,1] generating graphics... => 70
[2,1,1,1,1,1] generating graphics... => 21
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 1
[7,1] generating graphics... => 8
[6,2] generating graphics... => 8
[6,1,1] generating graphics... => 28
[5,3] generating graphics... => 8
[5,2,1] generating graphics... => 56
[5,1,1,1] generating graphics... => 56
[4,4] generating graphics... => 4
[4,3,1] generating graphics... => 56
[4,2,2] generating graphics... => 28
[4,2,1,1] generating graphics... => 168
[4,1,1,1,1] generating graphics... => 70
[3,3,2] generating graphics... => 28
[3,3,1,1] generating graphics... => 84
[3,2,2,1] generating graphics... => 168
[3,2,1,1,1] generating graphics... => 280
[3,1,1,1,1,1] generating graphics... => 56
[2,2,2,2] generating graphics... => 14
[2,2,2,1,1] generating graphics... => 140
[2,2,1,1,1,1] generating graphics... => 140
[2,1,1,1,1,1,1] generating graphics... => 28
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 1
[8,1] generating graphics... => 9
[7,2] generating graphics... => 9
[7,1,1] generating graphics... => 36
[6,3] generating graphics... => 9
[6,2,1] generating graphics... => 72
[6,1,1,1] generating graphics... => 84
[5,4] generating graphics... => 9
[5,3,1] generating graphics... => 72
[5,2,2] generating graphics... => 36
[5,2,1,1] generating graphics... => 252
[5,1,1,1,1] generating graphics... => 126
[4,4,1] generating graphics... => 36
[4,3,2] generating graphics... => 72
[4,3,1,1] generating graphics... => 252
[4,2,2,1] generating graphics... => 252
[4,2,1,1,1] generating graphics... => 504
[4,1,1,1,1,1] generating graphics... => 126
[3,3,3] generating graphics... => 12
[3,3,2,1] generating graphics... => 252
[3,3,1,1,1] generating graphics... => 252
[3,2,2,2] generating graphics... => 84
[3,2,2,1,1] generating graphics... => 756
[3,2,1,1,1,1] generating graphics... => 630
[3,1,1,1,1,1,1] generating graphics... => 84
[2,2,2,2,1] generating graphics... => 126
[2,2,2,1,1,1] generating graphics... => 420
[2,2,1,1,1,1,1] generating graphics... => 252
[2,1,1,1,1,1,1,1] generating graphics... => 36
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 1
[9,1] generating graphics... => 10
[8,2] generating graphics... => 10
[8,1,1] generating graphics... => 45
[7,3] generating graphics... => 10
[7,2,1] generating graphics... => 90
[7,1,1,1] generating graphics... => 120
[6,4] generating graphics... => 10
[6,3,1] generating graphics... => 90
[6,2,2] generating graphics... => 45
[6,2,1,1] generating graphics... => 360
[6,1,1,1,1] generating graphics... => 210
[5,5] generating graphics... => 5
[5,4,1] generating graphics... => 90
[5,3,2] generating graphics... => 90
[5,3,1,1] generating graphics... => 360
[5,2,2,1] generating graphics... => 360
[5,2,1,1,1] generating graphics... => 840
[5,1,1,1,1,1] generating graphics... => 252
[4,4,2] generating graphics... => 45
[4,4,1,1] generating graphics... => 180
[4,3,3] generating graphics... => 45
[4,3,2,1] generating graphics... => 720
[4,3,1,1,1] generating graphics... => 840
[4,2,2,2] generating graphics... => 120
[4,2,2,1,1] generating graphics... => 1260
[4,2,1,1,1,1] generating graphics... => 1260
[4,1,1,1,1,1,1] generating graphics... => 210
[3,3,3,1] generating graphics... => 120
[3,3,2,2] generating graphics... => 180
[3,3,2,1,1] generating graphics... => 1260
[3,3,1,1,1,1] generating graphics... => 630
[3,2,2,2,1] generating graphics... => 840
[3,2,2,1,1,1] generating graphics... => 2520
[3,2,1,1,1,1,1] generating graphics... => 1260
[3,1,1,1,1,1,1,1] generating graphics... => 120
[2,2,2,2,2] generating graphics... => 42
[2,2,2,2,1,1] generating graphics... => 630
[2,2,2,1,1,1,1] generating graphics... => 1050
[2,2,1,1,1,1,1,1] generating graphics... => 420
[2,1,1,1,1,1,1,1,1] generating graphics... => 45
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 1
[10,1] generating graphics... => 11
[9,2] generating graphics... => 11
[9,1,1] generating graphics... => 55
[8,3] generating graphics... => 11
[8,2,1] generating graphics... => 110
[8,1,1,1] generating graphics... => 165
[7,4] generating graphics... => 11
[7,3,1] generating graphics... => 110
[7,2,2] generating graphics... => 55
[7,2,1,1] generating graphics... => 495
[7,1,1,1,1] generating graphics... => 330
[6,5] generating graphics... => 11
[6,4,1] generating graphics... => 110
[6,3,2] generating graphics... => 110
[6,3,1,1] generating graphics... => 495
[6,2,2,1] generating graphics... => 495
[6,2,1,1,1] generating graphics... => 1320
[6,1,1,1,1,1] generating graphics... => 462
[5,5,1] generating graphics... => 55
[5,4,2] generating graphics... => 110
[5,4,1,1] generating graphics... => 495
[5,3,3] generating graphics... => 55
[5,3,2,1] generating graphics... => 990
[5,3,1,1,1] generating graphics... => 1320
[5,2,2,2] generating graphics... => 165
[5,2,2,1,1] generating graphics... => 1980
[5,2,1,1,1,1] generating graphics... => 2310
[5,1,1,1,1,1,1] generating graphics... => 462
[4,4,3] generating graphics... => 55
[4,4,2,1] generating graphics... => 495
[4,4,1,1,1] generating graphics... => 660
[4,3,3,1] generating graphics... => 495
[4,3,2,2] generating graphics... => 495
[4,3,2,1,1] generating graphics... => 3960
[4,3,1,1,1,1] generating graphics... => 2310
[4,2,2,2,1] generating graphics... => 1320
[4,2,2,1,1,1] generating graphics... => 4620
[4,2,1,1,1,1,1] generating graphics... => 2772
[4,1,1,1,1,1,1,1] generating graphics... => 330
[3,3,3,2] generating graphics... => 165
[3,3,3,1,1] generating graphics... => 660
[3,3,2,2,1] generating graphics... => 1980
[3,3,2,1,1,1] generating graphics... => 4620
[3,3,1,1,1,1,1] generating graphics... => 1386
[3,2,2,2,2] generating graphics... => 330
[3,2,2,2,1,1] generating graphics... => 4620
[3,2,2,1,1,1,1] generating graphics... => 6930
[3,2,1,1,1,1,1,1] generating graphics... => 2310
[3,1,1,1,1,1,1,1,1] generating graphics... => 165
[2,2,2,2,2,1] generating graphics... => 462
[2,2,2,2,1,1,1] generating graphics... => 2310
[2,2,2,1,1,1,1,1] generating graphics... => 2310
[2,2,1,1,1,1,1,1,1] generating graphics... => 660
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 55
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 1
[11,1] generating graphics... => 12
[10,2] generating graphics... => 12
[10,1,1] generating graphics... => 66
[9,3] generating graphics... => 12
[9,2,1] generating graphics... => 132
[9,1,1,1] generating graphics... => 220
[8,4] generating graphics... => 12
[8,3,1] generating graphics... => 132
[8,2,2] generating graphics... => 66
[8,2,1,1] generating graphics... => 660
[8,1,1,1,1] generating graphics... => 495
[7,5] generating graphics... => 12
[7,4,1] generating graphics... => 132
[7,3,2] generating graphics... => 132
[7,3,1,1] generating graphics... => 660
[7,2,2,1] generating graphics... => 660
[7,2,1,1,1] generating graphics... => 1980
[7,1,1,1,1,1] generating graphics... => 792
[6,6] generating graphics... => 6
[6,5,1] generating graphics... => 132
[6,4,2] generating graphics... => 132
[6,4,1,1] generating graphics... => 660
[6,3,3] generating graphics... => 66
[6,3,2,1] generating graphics... => 1320
[6,3,1,1,1] generating graphics... => 1980
[6,2,2,2] generating graphics... => 220
[6,2,2,1,1] generating graphics... => 2970
[6,2,1,1,1,1] generating graphics... => 3960
[6,1,1,1,1,1,1] generating graphics... => 924
[5,5,2] generating graphics... => 66
[5,5,1,1] generating graphics... => 330
[5,4,3] generating graphics... => 132
[5,4,2,1] generating graphics... => 1320
[5,4,1,1,1] generating graphics... => 1980
[5,3,3,1] generating graphics... => 660
[5,3,2,2] generating graphics... => 660
[5,3,2,1,1] generating graphics... => 5940
[5,3,1,1,1,1] generating graphics... => 3960
[5,2,2,2,1] generating graphics... => 1980
[5,2,2,1,1,1] generating graphics... => 7920
[5,2,1,1,1,1,1] generating graphics... => 5544
[5,1,1,1,1,1,1,1] generating graphics... => 792
[4,4,4] generating graphics... => 22
[4,4,3,1] generating graphics... => 660
[4,4,2,2] generating graphics... => 330
[4,4,2,1,1] generating graphics... => 2970
[4,4,1,1,1,1] generating graphics... => 1980
[4,3,3,2] generating graphics... => 660
[4,3,3,1,1] generating graphics... => 2970
[4,3,2,2,1] generating graphics... => 5940
[4,3,2,1,1,1] generating graphics... => 15840
[4,3,1,1,1,1,1] generating graphics... => 5544
[4,2,2,2,2] generating graphics... => 495
[4,2,2,2,1,1] generating graphics... => 7920
[4,2,2,1,1,1,1] generating graphics... => 13860
[4,2,1,1,1,1,1,1] generating graphics... => 5544
[4,1,1,1,1,1,1,1,1] generating graphics... => 495
[3,3,3,3] generating graphics... => 55
[3,3,3,2,1] generating graphics... => 1980
[3,3,3,1,1,1] generating graphics... => 2640
[3,3,2,2,2] generating graphics... => 990
[3,3,2,2,1,1] generating graphics... => 11880
[3,3,2,1,1,1,1] generating graphics... => 13860
[3,3,1,1,1,1,1,1] generating graphics... => 2772
[3,2,2,2,2,1] generating graphics... => 3960
[3,2,2,2,1,1,1] generating graphics... => 18480
[3,2,2,1,1,1,1,1] generating graphics... => 16632
[3,2,1,1,1,1,1,1,1] generating graphics... => 3960
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 220
[2,2,2,2,2,2] generating graphics... => 132
[2,2,2,2,2,1,1] generating graphics... => 2772
[2,2,2,2,1,1,1,1] generating graphics... => 6930
[2,2,2,1,1,1,1,1,1] generating graphics... => 4620
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 990
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 66
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
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)

Created
May 31, 2016 at 14:57 by Christian Stump
Updated
Oct 29, 2017 at 21:33 by Martin Rubey