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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000749
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[.,.],.]
=> [1,2] => [1,1]
=> [1]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1]
=> [1]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [2,2]
=> [2]
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> [1]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,2]
=> [2]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2]
=> [2]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,2]
=> [2]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,2,1]
=> [2,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2]
=> [2]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,1]
=> [1]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,2]
=> [2]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,2]
=> [2]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2]
=> [2]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,1]
=> [1]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1]
=> [1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,1]
=> [1]
=> 0
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [4,2]
=> [2]
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [4,2]
=> [2]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [4,2]
=> [2]
=> 0
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St000326
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,1,0,0]
=> 1100 => 1001 => 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 101010 => 111111 => 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 110100 => 100001 => 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 111000 => 101101 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1111111111 => ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1111111001 => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1111100111 => ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1111101101 => ? = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1110011111 => ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1110000111 => ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1110110111 => ? = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1110000001 => ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1110110001 => ? = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1110111101 => ? = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001111111 => ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1001111001 => ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1001100001 => ? = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1000011001 => ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1011011111 => ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1000000111 => ? = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1000110111 => ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1011110111 => ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ? = 1 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1000001101 => ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1000110001 => ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1000111101 => ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1000100101 => ? = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1011000001 => ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 1011001101 => ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1011110001 => ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1010010001 => ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1011111101 => 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 1011100101 => ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1010011101 => ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1010000101 => ? = 0 + 1
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 1010110101 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 111111111111 => ? = 2 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 111111111001 => ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => 111111100111 => ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 111111101101 => ? = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => 111110011111 => ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => 111110011001 => ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => 111110000111 => ? = 1 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => 111110110111 => ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 111110000001 => 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 111110001101 => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 111110111101 => ? = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => 111001111111 => ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => 111001100111 => ? = 2 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 101100110100 => 111001100001 => 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 101101001010 => 111000011111 => ? = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 101101001100 => 111000011001 => 1 = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 101110001010 => 111011011111 => ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 101101010010 => 111000000111 => ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 101101100010 => 111000110111 => ? = 1 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 101110010010 => 111011000111 => ? = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 101110100010 => 111011110111 => ? = 0 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => 111010010111 => ? = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 101101100100 => 111000110001 => 1 = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 101101110000 => 111000100101 => 1 = 0 + 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => 111011000001 => 1 = 0 + 1
[.,[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 111010010001 => 1 = 0 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 111010000101 => 1 = 0 + 1
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 110010110100 => 100111100001 => 1 = 0 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => 100110011001 => 1 = 0 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 110011100100 => 100110110001 => 1 = 0 + 1
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => 100110100101 => 1 = 0 + 1
[[[.,.],.],[[.,[.,.]],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 111000110100 => 101101100001 => 1 = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 111001001100 => 101100011001 => 1 = 0 + 1
[[[[.,.],.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => 101001011001 => 1 = 0 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 111001100100 => 101100110001 => 1 = 0 + 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 111001110000 => 101100100101 => 1 = 0 + 1
[[[.,[.,.]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => 101111000001 => 1 = 0 + 1
[[[.,[[.,.],.]],[.,.]],.]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => 101110010001 => 1 = 0 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 111100100100 => 101001110001 => 1 = 0 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 111110000100 => 101011010001 => 1 = 0 + 1
[[[.,[[.,[.,.]],.]],.],.]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 111011010000 => 101110000101 => 1 = 0 + 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 111100110000 => 101001100101 => 1 = 0 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 111110010000 => 101011000101 => 1 = 0 + 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => 101010010101 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St001719
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,4,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 1 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,4,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [4,3,2,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [4,3,2,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => [5,4,6,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [6,5,3,2,4,1] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [5,6,3,2,4,1] => [5,3,6,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [6,5,2,3,4,1] => [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [6,4,3,2,5,1] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 0 + 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [5,4,2,3,6,1] => [5,2,4,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> [6,5,3,4,1,2] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> [6,4,3,5,1,2] => [4,6,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> [5,3,4,6,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[.,[[.,.],.]],[[.,.],.]]
=> [5,6,2,3,1,4] => [2,5,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],[.,.]]
=> [6,3,4,2,1,5] => [3,6,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 1
[[.,[[.,.],[[.,.],.]]],.]
=> [4,5,2,3,1,6] => [4,5,2,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> [5,3,4,1,2,6] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [6,2,5,7,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [5,3,6,2,7,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [7,4,5,6,3,1,2] => [4,7,5,1,3,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [6,4,5,3,7,1,2] => [4,6,1,5,3,7,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [6,5,3,4,7,1,2] => [3,6,1,5,7,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],[[.,.],.]]
=> [6,7,3,4,2,1,5] => [3,6,4,2,7,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],[[.,.],.]]
=> [6,7,4,2,3,1,5] => [2,6,4,7,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [3,5,7,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001720
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 0 + 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 1 + 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 2
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,4,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 2
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [4,3,2,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,11),(4,14),(5,12),(5,15),(6,13),(6,15),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,7),(15,9),(15,10)],16)
=> ? = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ? = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 1 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,4,3,6,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? = 0 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [4,3,2,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? = 1 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [4,3,2,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => [5,4,6,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? = 0 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> [6,5,3,2,4,1] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ? = 1 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> [5,6,3,2,4,1] => [5,3,6,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[.,[[[.,.],.],[.,[.,.]]]]
=> [6,5,2,3,4,1] => [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ? = 0 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> [6,4,3,2,5,1] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ? = 0 + 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [5,4,2,3,6,1] => [5,2,4,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 0 + 2
[[.,.],[[.,.],[.,[.,.]]]]
=> [6,5,3,4,1,2] => [3,6,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[.,.],[[.,[.,.]],[.,.]]]
=> [6,4,3,5,1,2] => [4,6,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[.,.],[[[.,.],[.,.]],.]]
=> [5,3,4,6,1,2] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[.,[[.,.],.]],[[.,.],.]]
=> [5,6,2,3,1,4] => [2,5,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[.,[.,[[.,.],.]]],[.,.]]
=> [6,3,4,2,1,5] => [3,6,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 0 + 2
[[.,[[.,.],[[.,.],.]]],.]
=> [4,5,2,3,1,6] => [4,5,2,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> [5,3,4,1,2,6] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [6,2,5,7,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [5,3,6,2,7,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [7,4,5,6,3,1,2] => [4,7,5,1,3,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[.,.],[[.,[[.,.],[.,.]]],.]]
=> [6,4,5,3,7,1,2] => [4,6,1,5,3,7,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [6,5,3,4,7,1,2] => [3,6,1,5,7,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[.,[.,[[.,.],.]]],[[.,.],.]]
=> [6,7,3,4,2,1,5] => [3,6,4,2,7,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[.,[[.,.],[.,.]]],[[.,.],.]]
=> [6,7,4,2,3,1,5] => [2,6,4,7,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => [3,5,7,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001771
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1,3] => [2,1,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => [2,3,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => [5,4,1,3,2,6] => [5,4,1,3,2,6] => ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [5,4,1,2,3,6] => [5,4,1,2,3,6] => ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [5,1,4,3,2,6] => [5,1,4,3,2,6] => ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => [5,2,1,4,3,6] => [5,2,1,4,3,6] => ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,3,2,1,4,6] => [5,3,2,1,4,6] => ? = 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [5,3,1,2,4,6] => [5,3,1,2,4,6] => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [5,2,1,3,4,6] => [5,2,1,3,4,6] => ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [2,6,5,4,3,1] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [2,4,6,5,3,1] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [2,5,4,6,3,1] => [1,5,3,2,4,6] => [1,5,3,2,4,6] => ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [3,2,6,5,4,1] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [3,2,5,6,4,1] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [4,3,2,6,5,1] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [3,4,2,6,5,1] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [2,4,3,6,5,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [3,2,4,6,5,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 0
[.,[[[[.,.],.],.],[.,.]]]
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [5,4,3,2,6,1] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [4,5,3,2,6,1] => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [3,5,4,2,6,1] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [3,4,5,2,6,1] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [2,5,4,3,6,1] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [3,2,5,4,6,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 1
[.,[[[[.,.],.],[.,.]],.]]
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1,3] => [2,1,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => [2,3,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => [5,4,1,3,2,6] => [5,4,1,3,2,6] => ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [5,4,1,2,3,6] => [5,4,1,2,3,6] => ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [5,1,4,3,2,6] => [5,1,4,3,2,6] => ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => [5,2,1,4,3,6] => [5,2,1,4,3,6] => ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,3,2,1,4,6] => [5,3,2,1,4,6] => ? = 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [5,3,1,2,4,6] => [5,3,1,2,4,6] => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [5,2,1,3,4,6] => [5,2,1,3,4,6] => ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [2,6,5,4,3,1] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [2,4,6,5,3,1] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [2,5,4,6,3,1] => [1,5,3,2,4,6] => [1,5,3,2,4,6] => ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [3,2,6,5,4,1] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [3,2,5,6,4,1] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [4,3,2,6,5,1] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [3,4,2,6,5,1] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [2,4,3,6,5,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [3,2,4,6,5,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 0
[.,[[[[.,.],.],.],[.,.]]]
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [5,4,3,2,6,1] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [4,5,3,2,6,1] => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [3,5,4,2,6,1] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [3,4,5,2,6,1] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [2,5,4,3,6,1] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [3,2,5,4,6,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 1
[.,[[[[.,.],.],[.,.]],.]]
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1,3] => [2,1,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => [2,3,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 1
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => [5,4,1,3,2,6] => [5,4,1,3,2,6] => ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [5,4,1,2,3,6] => [5,4,1,2,3,6] => ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [5,1,4,3,2,6] => [5,1,4,3,2,6] => ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => [5,2,1,4,3,6] => [5,2,1,4,3,6] => ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,3,2,1,4,6] => [5,3,2,1,4,6] => ? = 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [5,3,1,2,4,6] => [5,3,1,2,4,6] => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [5,2,1,3,4,6] => [5,2,1,3,4,6] => ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [2,6,5,4,3,1] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [2,4,6,5,3,1] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [2,5,4,6,3,1] => [1,5,3,2,4,6] => [1,5,3,2,4,6] => ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [3,2,6,5,4,1] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [3,2,5,6,4,1] => [2,1,5,3,4,6] => [2,1,5,3,4,6] => ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [4,3,2,6,5,1] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [3,4,2,6,5,1] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [2,4,3,6,5,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [3,2,4,6,5,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 0
[.,[[[[.,.],.],.],[.,.]]]
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [5,4,3,2,6,1] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [4,5,3,2,6,1] => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [3,5,4,2,6,1] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [3,4,5,2,6,1] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [2,5,4,3,6,1] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [3,2,5,4,6,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 1
[.,[[[[.,.],.],[.,.]],.]]
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St000068
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 1 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? = 2 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [4,3,6,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(1,18),(1,21),(2,11),(2,13),(2,15),(3,12),(3,14),(3,15),(4,8),(4,10),(4,12),(4,15),(5,9),(5,10),(5,13),(5,15),(6,1),(6,8),(6,9),(6,11),(6,14),(8,17),(8,18),(8,19),(8,20),(9,17),(9,18),(9,19),(9,20),(10,20),(10,21),(11,17),(11,19),(12,18),(12,20),(13,17),(13,21),(14,18),(14,19),(14,21),(15,19),(15,20),(15,21),(16,7),(17,16),(17,22),(18,16),(18,22),(19,16),(19,22),(20,16),(20,22),(21,22),(22,7)],23)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [5,2,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(1,8),(1,18),(2,11),(2,13),(2,14),(2,18),(3,10),(3,12),(3,14),(3,18),(4,8),(4,9),(4,12),(4,13),(5,6),(5,9),(5,10),(5,11),(6,23),(8,19),(8,23),(9,16),(9,17),(9,23),(10,16),(10,20),(10,23),(11,17),(11,20),(11,23),(12,15),(12,16),(12,19),(13,15),(13,17),(13,19),(14,15),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21)],24)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [5,2,6,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(1,18),(1,21),(2,11),(2,13),(2,15),(3,12),(3,14),(3,15),(4,8),(4,10),(4,12),(4,15),(5,9),(5,10),(5,13),(5,15),(6,1),(6,8),(6,9),(6,11),(6,14),(8,17),(8,18),(8,19),(8,20),(9,17),(9,18),(9,19),(9,20),(10,20),(10,21),(11,17),(11,19),(12,18),(12,20),(13,17),(13,21),(14,18),(14,19),(14,21),(15,19),(15,20),(15,21),(16,7),(17,16),(17,22),(18,16),(18,22),(19,16),(19,22),(20,16),(20,22),(21,22),(22,7)],23)
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [3,4,5,2,6,1] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [3,4,6,1,5,2] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [3,5,2,4,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 1 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [5,2,3,4,6,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [3,6,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [3,5,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [5,2,3,6,1,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,24),(1,25),(2,9),(2,11),(2,13),(2,15),(3,8),(3,10),(3,13),(3,14),(4,8),(4,11),(4,12),(4,16),(5,9),(5,10),(5,12),(5,17),(6,1),(6,14),(6,15),(6,16),(6,17),(8,20),(8,24),(9,20),(9,25),(10,20),(10,23),(10,25),(11,20),(11,23),(11,24),(12,19),(12,20),(13,18),(13,24),(13,25),(14,18),(14,23),(14,24),(15,18),(15,23),(15,25),(16,19),(16,23),(16,24),(16,25),(17,19),(17,23),(17,24),(17,25),(18,22),(19,21),(19,22),(20,21),(21,7),(22,7),(23,21),(23,22),(24,21),(24,22),(25,21),(25,22)],26)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [5,3,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 2 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,13),(1,16),(1,17),(2,9),(2,13),(2,15),(2,17),(3,12),(3,14),(3,15),(3,17),(4,11),(4,14),(4,16),(4,17),(5,8),(5,11),(5,12),(5,15),(5,16),(6,8),(6,9),(6,10),(6,15),(6,16),(8,19),(8,20),(9,19),(9,22),(9,23),(10,20),(10,22),(10,23),(11,19),(11,22),(11,24),(12,20),(12,22),(12,24),(13,22),(13,23),(14,22),(14,24),(15,19),(15,20),(15,23),(15,24),(16,19),(16,20),(16,23),(16,24),(17,23),(17,24),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21),(23,18),(23,21),(24,18),(24,21)],25)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [6,5,3,2,4,1] => [3,5,4,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [5,6,3,2,4,1] => [3,6,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,18),(1,19),(2,7),(2,13),(2,15),(3,7),(3,12),(3,15),(4,8),(4,9),(4,12),(4,15),(5,8),(5,10),(5,13),(5,15),(6,1),(6,9),(6,10),(6,12),(6,13),(7,17),(8,14),(8,21),(9,14),(9,18),(9,19),(9,21),(10,14),(10,18),(10,19),(10,21),(12,17),(12,18),(12,21),(13,17),(13,19),(13,21),(14,16),(14,20),(15,17),(15,21),(16,11),(17,20),(18,16),(18,20),(19,16),(19,20),(20,11),(21,16),(21,20)],22)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [6,5,2,3,4,1] => [5,4,3,2,6,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [6,4,3,2,5,1] => [3,4,2,5,6,1] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [6,3,4,2,5,1] => [4,2,3,5,6,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [6,4,2,3,5,1] => [4,3,2,5,6,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [6,3,2,4,5,1] => [3,2,4,5,6,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 0 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => [5,4,3,2,1,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
[[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001845
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> ?
=> ? = 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ?
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ?
=> ? = 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ?
=> ? = 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? = 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? = 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ?
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? = 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? = 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? = 0
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? = 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 0
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => ([],6)
=> ?
=> ? = 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => ([(4,5)],6)
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => ([(4,5)],6)
=> ?
=> ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => ([(4,5)],6)
=> ?
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> ?
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
=> ?
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> ?
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6)
=> ?
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => ([(4,5)],6)
=> ?
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> ?
=> ? = 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6)
=> ?
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [6,5,3,2,4,1] => ([(3,5),(4,5)],6)
=> ?
=> ? = 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6)
=> ?
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [6,5,2,3,4,1] => ([(3,4),(4,5)],6)
=> ?
=> ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 1
[[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 0
[[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 0
[[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 0
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 0
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
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