Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001099
Mapping
St001099: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 2
[1,1,1]
 => 3
[4]
 => 0
[3,1]
 => 6
[2,2]
 => 10
[2,1,1]
 => 12
[1,1,1,1]
 => 15
[5]
 => 0
[4,1]
 => 24
[3,2]
 => 54
[3,1,1]
 => 60
[2,2,1]
 => 78
[2,1,1,1]
 => 90
[1,1,1,1,1]
 => 105
[6]
 => 0
[5,1]
 => 120
[4,2]
 => 336
[4,1,1]
 => 360
[3,3]
 => 450
[3,2,1]
 => 570
[3,1,1,1]
 => 630
[2,2,2]
 => 672
[2,2,1,1]
 => 750
[2,1,1,1,1]
 => 840
[1,1,1,1,1,1]
 => 945
[7]
 => 0
[6,1]
 => 720
[5,2]
 => 2400
[5,1,1]
 => 2520
[4,3]
 => 3960
[4,2,1]
 => 4680
[4,1,1,1]
 => 5040
[3,3,1]
 => 5670
[3,2,2]
 => 6360
[3,2,1,1]
 => 6930
[3,1,1,1,1]
 => 7560
[2,2,2,1]
 => 7860
[2,2,1,1,1]
 => 8610
[2,1,1,1,1,1]
 => 9450
[1,1,1,1,1,1,1]
 => 10395
[8]
 => 0
[7,1]
 => 5040
[6,2]
 => 19440
[6,1,1]
 => 20160
[5,3]
 => 37800
[5,2,1]
 => 42840
[5,1,1,1]
 => 45360
Description
The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1]. This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2] Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.