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Matching statistic: St000517
Mapping
St000517: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
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 $\frac{1}{n+1}\binom{2n}{n}$.