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Identifier

St001090: Permutations ⟶ ℤ

Values

[] => 0

[1] => 0

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 1

[1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + 2\ q^{2}$

$F_{4} = 1 + 7\ q + 8\ q^{2} + 8\ q^{3}$

$F_{5} = 1 + 15\ q + 26\ q^{2} + 46\ q^{3} + 32\ q^{4}$

$F_{6} = 1 + 31\ q + 80\ q^{2} + 191\ q^{3} + 262\ q^{4} + 155\ q^{5}$

Description

The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation

$\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is

$t$ -pop-stack sortable if it is sortable using

$t$ pop-stacks in series.

References

[1] Claesson, A., Ágúst Gu\eth mundsson, B. Enumerating permutations sortable by $k$ passes through a pop-stack arXiv:1710.04978
[2] Pudwell, L., Smith, R. Two-stack-sorting with pop stacks arXiv:1801.05005
[3] Elder, M., Goh, Y. K. $k$ -pop stack sortable permutations and $2$ -avoidance arXiv:1911.03104

Code

def pop_stack_sort(x):
    """
    sage: pop_stack_sort([7,5,2,4,9,1,8,6,3])
    [2, 5, 7, 4, 1, 9, 3, 6, 8]
    """
    stack = []
    result = []
    for e in x:
        while stack and e > stack[-1]:
            result += stack[::-1]
            stack = []
        stack.append(e)
        
    result += stack[::-1]
    return Permutation(result)

def statistic(x):
    i = 0
    id = Permutations(x.size()).identity()
    while x != id:
        x = pop_stack_sort(x)
        i += 1
    return i

Created

Jan 17, 2018 at 09:27 by Martin Rubey

Updated

Nov 17, 2020 at 00:56 by Travis Scrimshaw