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Identifier

St001262: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 4

[1,1] => 3

[3] => 9

[2,1] => 7

[1,1,1] => 6

[4] => 16

[3,1] => 13

[2,2] => 12

[2,1,1] => 11

[1,1,1,1] => 10

[5] => 25

[4,1] => 21

[3,2] => 19

[3,1,1] => 18

[2,2,1] => 17

[2,1,1,1] => 16

[1,1,1,1,1] => 15

[6] => 36

[5,1] => 31

[4,2] => 28

[4,1,1] => 27

[3,3] => 27

[3,2,1] => 25

[3,1,1,1] => 24

[2,2,2] => 24

[2,2,1,1] => 23

[2,1,1,1,1] => 22

[1,1,1,1,1,1] => 21

[7] => 49

[6,1] => 43

[5,2] => 39

[5,1,1] => 38

[4,3] => 37

[4,2,1] => 35

[4,1,1,1] => 34

[3,3,1] => 34

[3,2,2] => 33

[3,2,1,1] => 32

[3,1,1,1,1] => 31

[2,2,2,1] => 31

[2,2,1,1,1] => 30

[2,1,1,1,1,1] => 29

[1,1,1,1,1,1,1] => 28

[8] => 64

[7,1] => 57

[6,2] => 52

[6,1,1] => 51

[5,3] => 49

[5,2,1] => 47

[5,1,1,1] => 46

[4,4] => 48

[4,3,1] => 45

[4,2,2] => 44

[4,2,1,1] => 43

[4,1,1,1,1] => 42

[3,3,2] => 43

[3,3,1,1] => 42

[3,2,2,1] => 41

[3,2,1,1,1] => 40

[3,1,1,1,1,1] => 39

[2,2,2,2] => 40

[2,2,2,1,1] => 39

[2,2,1,1,1,1] => 38

[2,1,1,1,1,1,1] => 37

[1,1,1,1,1,1,1,1] => 36

[9] => 81

[8,1] => 73

[7,2] => 67

[7,1,1] => 66

[6,3] => 63

[6,2,1] => 61

[6,1,1,1] => 60

[5,4] => 61

[5,3,1] => 58

[5,2,2] => 57

[5,2,1,1] => 56

[5,1,1,1,1] => 55

[4,4,1] => 57

[4,3,2] => 55

[4,3,1,1] => 54

[4,2,2,1] => 53

[4,2,1,1,1] => 52

[4,1,1,1,1,1] => 51

[3,3,3] => 54

[3,3,2,1] => 52

[3,3,1,1,1] => 51

[3,2,2,2] => 51

[3,2,2,1,1] => 50

[3,2,1,1,1,1] => 49

[3,1,1,1,1,1,1] => 48

[2,2,2,2,1] => 49

[2,2,2,1,1,1] => 48

[2,2,1,1,1,1,1] => 47

[2,1,1,1,1,1,1,1] => 46

[1,1,1,1,1,1,1,1,1] => 45

[10] => 100

[9,1] => 91

[8,2] => 84

[8,1,1] => 83

[7,3] => 79

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Description

The dimension of the maximal parabolic seaweed algebra corresponding to the partition.
Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be two compositions of $n$. The corresponding seaweed algebra is the associative subalgebra of the algebra of $n\times n$ matrices which preserves the flags
$$ \{0\} \subset V_1 \subset \cdots \subset V_{m-1} \subset V_m =V $$
and
$$ V=W_0\supset W_1\supset \cdots \supset W_t=\{0\}, $$
where $V_i=\text{span}\{e_1,\dots, e_{a_1+\cdots +a_i}\}$ and $W_j=\text{span}\{e_{b_1+\cdots +b_j+1},\dots, e_n\}$.
Thus, its dimension is
$$ \frac{1}{2}\left(\sum a_i^2 + \sum b_i^2\right). $$
It is maximal parabolic if $b_1=n$.

References

[1] Dergachev, V., Kirillov, A. Index of Lie algebras of seaweed type MathSciNet:1774864
[2] Coll, V., Mayers, A., Mayers, N. Statistics on integer partitions arising from seaweed algebras arXiv:1809.09271

Code

def statistic(la):
    return sum(p^2/2 for p in la) + la.size()^2/2

Created

Sep 26, 2018 at 05:12 by Martin Rubey

Updated

May 12, 2020 at 15:38 by Martin Rubey