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Identifier

Mp00204: Permutations —LLPS⟶ Integer partitions

Images

[] => []

[1] => [1]

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

[2,1] => [2]

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

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

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

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

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

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

[1,2,3,4] => [1,1,1,1]

[1,2,4,3] => [2,1,1]

[1,3,2,4] => [2,1,1]

[1,3,4,2] => [2,1,1]

[1,4,2,3] => [2,1,1]

[1,4,3,2] => [3,1]

[2,1,3,4] => [2,1,1]

[2,1,4,3] => [2,2]

[2,3,1,4] => [2,1,1]

[2,3,4,1] => [2,1,1]

[2,4,1,3] => [2,1,1]

[2,4,3,1] => [3,1]

[3,1,2,4] => [2,1,1]

[3,1,4,2] => [2,2]

[3,2,1,4] => [3,1]

[3,2,4,1] => [3,1]

[3,4,1,2] => [2,1,1]

[3,4,2,1] => [3,1]

[4,1,2,3] => [2,1,1]

[4,1,3,2] => [3,1]

[4,2,1,3] => [3,1]

[4,2,3,1] => [3,1]

[4,3,1,2] => [3,1]

[4,3,2,1] => [4]

[1,2,3,4,5] => [1,1,1,1,1]

[1,2,3,5,4] => [2,1,1,1]

[1,2,4,3,5] => [2,1,1,1]

[1,2,4,5,3] => [2,1,1,1]

[1,2,5,3,4] => [2,1,1,1]

[1,2,5,4,3] => [3,1,1]

[1,3,2,4,5] => [2,1,1,1]

[1,3,2,5,4] => [2,2,1]

[1,3,4,2,5] => [2,1,1,1]

[1,3,4,5,2] => [2,1,1,1]

[1,3,5,2,4] => [2,1,1,1]

[1,3,5,4,2] => [3,1,1]

[1,4,2,3,5] => [2,1,1,1]

[1,4,2,5,3] => [2,2,1]

[1,4,3,2,5] => [3,1,1]

[1,4,3,5,2] => [3,1,1]

[1,4,5,2,3] => [2,1,1,1]

[1,4,5,3,2] => [3,1,1]

[1,5,2,3,4] => [2,1,1,1]

[1,5,2,4,3] => [3,1,1]

[1,5,3,2,4] => [3,1,1]

[1,5,3,4,2] => [3,1,1]

[1,5,4,2,3] => [3,1,1]

[1,5,4,3,2] => [4,1]

[2,1,3,4,5] => [2,1,1,1]

[2,1,3,5,4] => [2,2,1]

[2,1,4,3,5] => [2,2,1]

[2,1,4,5,3] => [2,2,1]

[2,1,5,3,4] => [2,2,1]

[2,1,5,4,3] => [3,2]

[2,3,1,4,5] => [2,1,1,1]

[2,3,1,5,4] => [2,2,1]

[2,3,4,1,5] => [2,1,1,1]

[2,3,4,5,1] => [2,1,1,1]

[2,3,5,1,4] => [2,1,1,1]

[2,3,5,4,1] => [3,1,1]

[2,4,1,3,5] => [2,1,1,1]

[2,4,1,5,3] => [2,2,1]

[2,4,3,1,5] => [3,1,1]

[2,4,3,5,1] => [3,1,1]

[2,4,5,1,3] => [2,1,1,1]

[2,4,5,3,1] => [3,1,1]

[2,5,1,3,4] => [2,1,1,1]

[2,5,1,4,3] => [3,1,1]

[2,5,3,1,4] => [3,1,1]

[2,5,3,4,1] => [3,1,1]

[2,5,4,1,3] => [3,1,1]

[2,5,4,3,1] => [4,1]

[3,1,2,4,5] => [2,1,1,1]

[3,1,2,5,4] => [2,2,1]

[3,1,4,2,5] => [2,2,1]

[3,1,4,5,2] => [2,2,1]

[3,1,5,2,4] => [2,2,1]

[3,1,5,4,2] => [3,2]

[3,2,1,4,5] => [3,1,1]

[3,2,1,5,4] => [3,2]

[3,2,4,1,5] => [3,1,1]

[3,2,4,5,1] => [3,1,1]

[3,2,5,1,4] => [3,1,1]

[3,2,5,4,1] => [3,2]

[3,4,1,2,5] => [2,1,1,1]

[3,4,1,5,2] => [2,2,1]

[3,4,2,1,5] => [3,1,1]

[3,4,2,5,1] => [3,1,1]

[3,4,5,1,2] => [2,1,1,1]

[3,4,5,2,1] => [3,1,1]

[3,5,1,2,4] => [2,1,1,1]

>>> Load all 40655 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

Description

The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence

$u = (u_1, u_2, \ldots)$ is an index

$i$ such that

$u_i < u_{i+1}$ . Let

$\mathrm{asc}(u)$ denote the number of ascents of

$u$ , and let

$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$
Given a permutation

$w$ in the symmetric group

$\mathfrak{S}_n$ , define

$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences

${u_i}$ of

$w$ .
Then

$A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of

$n$ . Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.

References

[1] Lewis, J., Lyu, H., Pylyavskyy, P., Sen, A. Scaling limit of soliton lengths in a multicolor box-ball system arXiv:1911.04458

Properties

surjective, graded

Sage code

def mapping(pi):
    to_standard = sage.combinat.permutation.to_standard
    def llis(u):
        if len(u) == 1:
            return 1
        if len(u) == 2:
            if u[0] < u[1]:
                return 2
            return 1
        if len(u) == 3:
            if u[0] < u[1] < u[2]:
                return 3
            if u[0] > u[1] > u[2]:
                return 1
            return 2

        return to_standard(u).longest_increasing_subsequence_length()

    la = []
    n = len(pi)
    t = 0
    pi = pi.reverse()
    for k in range(1, n+1):
        a = 0
        for p in Compositions(n, length=k):
            b = [sum(p[:i]) for i in range(k+1)]
            s = sum(llis(pi[b[i]:b[i+1]]) for i in range(k))
            a = max(a, s)
        d = a - t
        if a == t:
            return Partition(la)
        la.append(d)
        t = a

    return Partition(la)

Weight

Created

Jan 28, 2021 at 12:18 by Martin Rubey

Updated

Feb 01, 2021 at 12:27 by Martin Rubey