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Identifier

St001263: Integer compositions ⟶ ℤ

Values

[1] => 0

[1,1] => 0

[2] => 0

[1,1,1] => 1

[1,2] => 0

[2,1] => 0

[3] => 1

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

[1,1,2] => 0

[1,2,1] => 1

[1,3] => 0

[2,1,1] => 0

[2,2] => 1

[3,1] => 0

[4] => 1

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

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

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

[1,1,3] => 1

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

[1,2,2] => 0

[1,3,1] => 2

[1,4] => 0

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

[2,1,2] => 2

[2,2,1] => 0

[2,3] => 0

[3,1,1] => 1

[3,2] => 0

[4,1] => 0

[5] => 2

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

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

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

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

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

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

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

[1,1,4] => 0

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

[1,2,1,2] => 0

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

[1,2,3] => 0

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

[1,3,2] => 0

[1,4,1] => 2

[1,5] => 0

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

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

[2,1,2,1] => 0

[2,1,3] => 0

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

[2,2,2] => 2

[2,3,1] => 0

[2,4] => 1

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

[3,1,2] => 0

[3,2,1] => 0

[3,3] => 2

[4,1,1] => 0

[4,2] => 1

[5,1] => 0

[6] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,5] => 1

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

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

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

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

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

[1,2,2,2] => 0

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

[1,2,4] => 2

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

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

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

[1,3,3] => 1

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

[1,4,2] => 0

[1,5,1] => 3

[1,6] => 0

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

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

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

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

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

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

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Description

The index of the maximal parabolic seaweed algebra associated with the composition.
Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows:
* place $n$ dots on a horizontal line
* subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line
* subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line
By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is
$$ \operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1, $$
where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander.
This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.

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 to_meander(la, mu):
    """
    Return the meander associated with two partitions as a graph.
    """
    la = Composition(la)
    mu = Composition(mu)
    n = la.size()
    assert mu.size() == n
    M = Graph(n)
    offset = 0
    for part in la:
        M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)])
        offset += part
    offset = 0
    for part in mu:
        M.add_edges([(offset + i, offset + part-i-1) for i in range(part//2)])
        offset += part
    return M

def seaweed_index(la, mu):
    M = to_meander(la, mu)
    index = -1
    for G in M.connected_components_subgraphs():
        if G.is_cycle():
            index += 2
        else:
            index += 1
    return index

def statistic(la):
    return seaweed_index(la, la.size())

Created

Sep 26, 2018 at 04:49 by Martin Rubey

Updated

Sep 26, 2018 at 10:46 by Martin Rubey