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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1,0]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[[[]]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 10 => 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 10 => 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000444
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[],[],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[[]],[],[],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6
[[[]],[],[],[],[[]],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 6
[[[]],[],[],[],[[],[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[[[]],[],[],[],[[[]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5
[[[]],[],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 6
[[[]],[],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001062
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> {{1}}
=> {{1}}
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1,2}}
=> 2
[[[]]]
=> [1,1,0,0]
=> {{1,2}}
=> {{1},{2}}
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1,2,3}}
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1},{2},{3}}
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,3,4,5},{2}}
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1,2,4,5},{3}}
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,4,5},{2,3}}
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1,2,3,5},{4}}
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1,2,5},{3,4}}
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,5},{2,3,4}}
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,5},{2},{3,4}}
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,5},{2,4},{3}}
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,5},{2,3},{4}}
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1,2},{3,4,5}}
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1},{2,3,4,5}}
=> 4
[[[]],[],[],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 6
[[[]],[],[],[],[[]],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4},{5},{6,7},{8}}
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 6
[[[]],[],[],[],[[],[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1,2},{3},{4},{5},{6,8},{7}}
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 5
[[[]],[],[],[],[[[]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4},{5},{6,7,8}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> ? = 5
[[[]],[],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> {{1,2,3,5,6,7},{4},{8}}
=> ? = 6
[[[]],[],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1,2},{3},{4},{5,8},{6},{7}}
=> {{1,5,6,7},{2,3,4},{8}}
=> ? = 4
[[[],[],[],[],[[],[]]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 5
[[[],[],[],[],[[[]]]]]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 5
[[[],[],[],[[],[],[]]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,5,8},{2},{3},{4},{6},{7}}
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 4
Description
The maximal size of a block of a set partition.
Matching statistic: St000308
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
Values
[[]]
=> [[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [[],[]]
=> [[.,.],.]
=> [1,2] => 2
[[[]]]
=> [[[]]]
=> [.,[.,.]]
=> [2,1] => 1
[[],[],[]]
=> [[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[[],[[]]]
=> [[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[]],[]]
=> [[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[[[],[]]]
=> [[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[[]]]]
=> [[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[[],[],[[]]]
=> [[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3
[[],[[]],[]]
=> [[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[[],[[],[]]]
=> [[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[[[]],[],[]]
=> [[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[[[]],[[]]]
=> [[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[[[]]],[]]
=> [[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[[],[],[]]]
=> [[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[[],[[]]]]
=> [[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[[[]],[]]]
=> [[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[[[[],[]]]]
=> [[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[[[[[]]]]]
=> [[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 5
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 4
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 4
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 3
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 4
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 3
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 3
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 3
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 2
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 2
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 4
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 3
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 3
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2
[[[]],[],[[],[]],[]]
=> [[],[[],[]],[],[[]]]
=> [[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => ? = 4
[[[],[]],[],[],[],[]]
=> [[],[],[],[],[[],[]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => ? = 5
[[[],[],[],[],[]],[]]
=> [[],[[],[],[],[],[]]]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ? = 5
[[[],[],[[],[]],[]]]
=> [[[],[[],[]],[],[]]]
=> [.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => ? = 4
[[[]],[],[],[],[],[],[]]
=> [[],[],[],[],[],[],[[]]]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => ? = 7
[[[]],[],[],[],[],[[]]]
=> [[[]],[],[],[],[],[[]]]
=> [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [8,2,1,3,4,5,6,7] => ? = 6
[[[]],[],[],[],[[]],[]]
=> [[],[[]],[],[],[],[[]]]
=> [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => ? = 6
[[[]],[],[],[],[[],[]]]
=> [[[],[]],[],[],[],[[]]]
=> [[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [8,2,3,1,4,5,6,7] => ? = 5
[[[]],[],[],[],[[[]]]]
=> [[[[]]],[],[],[],[[]]]
=> [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [8,3,2,1,4,5,6,7] => ? = 5
[[[]],[],[],[[]],[],[]]
=> [[],[],[[]],[],[],[[]]]
=> [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => ? = 6
[[[]],[],[],[[],[],[]]]
=> [[[],[],[]],[],[],[[]]]
=> [[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [8,2,3,4,1,5,6,7] => ? = 4
[[[],[],[],[],[],[],[]]]
=> [[[],[],[],[],[],[],[]]]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => ? = 7
[[[],[],[],[],[],[[]]]]
=> [[[[]],[],[],[],[],[]]]
=> [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => ? = 6
[[[],[],[],[],[[]],[]]]
=> [[[],[[]],[],[],[],[]]]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => ? = 6
[[[],[],[],[],[[],[]]]]
=> [[[[],[]],[],[],[],[]]]
=> [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => ? = 5
[[[],[],[],[],[[[]]]]]
=> [[[[[]]],[],[],[],[]]]
=> [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => ? = 5
[[[],[],[],[[]],[],[]]]
=> [[[],[],[[]],[],[],[]]]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => ? = 6
[[[],[],[],[[],[],[]]]]
=> [[[[],[],[]],[],[],[]]]
=> [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => ? = 4
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000485
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 87%●distinct values known / distinct values provided: 86%
Values
[[]]
=> [.,.]
=> [1] => [1] => ? = 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [2,3,4,1] => 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,4,3,1] => 3
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,2,4,1] => 3
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => 2
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,4,2] => 3
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 3
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [2,3,4,5,1] => 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,3,5,4,1] => 4
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,4,3,5,1] => 4
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,5,4,3,1] => 3
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [2,5,3,4,1] => 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,2,4,5,1] => 4
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,2,5,4,1] => 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,2,5,1] => 3
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [4,2,3,5,1] => 3
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,4,2,1] => 3
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => 2
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => 2
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,3,4,5,2] => 4
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,3,5,4,2] => 3
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,4,3,5,2] => 3
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,5,3] => 3
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,5,3] => 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,3,1,5,4] => 3
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,3,4,1,5] => 4
[[[]],[[],[],[],[]]]
=> [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => [1,7,4,5,6,3,2] => ? = 4
[[[],[]],[],[],[],[]]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 5
[[[],[],[],[],[]],[]]
=> [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => [2,3,4,5,1,7,6] => ? = 5
[[[],[],[],[],[],[]]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => ? = 6
[[[],[],[],[],[[]]]]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [2,3,4,6,5,1,7] => ? = 5
[[[],[],[],[[]],[]]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [4,6,5,3,2,1,7] => [2,3,5,4,6,1,7] => ? = 5
[[[],[],[],[[],[]]]]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [2,3,6,5,4,1,7] => ? = 4
[[[],[],[],[[[]]]]]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => [2,3,6,4,5,1,7] => ? = 4
[[[],[],[[]],[],[]]]
=> [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => [2,4,3,5,6,1,7] => ? = 5
[[[],[],[[]],[[]]]]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => [2,4,3,6,5,1,7] => ? = 4
[[[],[],[[],[]],[]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => [2,5,4,3,6,1,7] => ? = 4
[[[],[],[[],[],[]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [2,6,4,5,3,1,7] => ? = 3
[[[],[],[[],[[]]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => [2,6,5,4,3,1,7] => ? = 3
[[[],[[]],[],[],[]]]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => [3,2,4,5,6,1,7] => ? = 5
[[[],[[],[],[],[]]]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [5,4,3,2,6,1,7] => [6,3,4,5,2,1,7] => ? = 4
[[[]],[],[],[],[],[],[]]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [1,3,4,5,6,7,8,2] => ? = 7
[[[]],[],[],[],[],[[]]]
=> [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [1,3,4,5,6,8,7,2] => ? = 6
[[[]],[],[],[],[[]],[]]
=> [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,6,8,7,5,4,3,2] => [1,3,4,5,7,6,8,2] => ? = 6
[[[]],[],[],[],[[],[]]]
=> [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,7,6,8,5,4,3,2] => [1,3,4,5,8,7,6,2] => ? = 5
[[[]],[],[],[],[[[]]]]
=> [[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,6,7,8,5,4,3,2] => [1,3,4,5,8,6,7,2] => ? = 5
[[[]],[],[],[[]],[],[]]
=> [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,5,8,7,6,4,3,2] => [1,3,4,6,5,7,8,2] => ? = 6
[[[]],[],[],[[],[],[]]]
=> [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,7,6,5,8,4,3,2] => [1,3,4,8,6,7,5,2] => ? = 4
[[[],[],[],[],[],[],[]]]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [2,3,4,5,6,7,1,8] => ? = 7
[[[],[],[],[],[],[[]]]]
=> [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [2,3,4,5,7,6,1,8] => ? = 6
[[[],[],[],[],[[]],[]]]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [5,7,6,4,3,2,1,8] => [2,3,4,6,5,7,1,8] => ? = 6
[[[],[],[],[],[[],[]]]]
=> [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [6,5,7,4,3,2,1,8] => [2,3,4,7,6,5,1,8] => ? = 5
[[[],[],[],[],[[[]]]]]
=> [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [5,6,7,4,3,2,1,8] => [2,3,4,7,5,6,1,8] => ? = 5
[[[],[],[],[[]],[],[]]]
=> [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [4,7,6,5,3,2,1,8] => [2,3,5,4,6,7,1,8] => ? = 6
[[[],[],[],[[],[],[]]]]
=> [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [6,5,4,7,3,2,1,8] => [2,3,7,5,6,4,1,8] => ? = 4
Description
The length of the longest cycle of a permutation.
Matching statistic: St000328
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St000328: Ordered trees ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 86%
Values
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 3
[[],[[]]]
=> 2
[[[]],[]]
=> 2
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 4
[[],[],[[]]]
=> 3
[[],[[]],[]]
=> 3
[[],[[],[]]]
=> 2
[[],[[[]]]]
=> 2
[[[]],[],[]]
=> 3
[[[]],[[]]]
=> 2
[[[],[]],[]]
=> 2
[[[[]]],[]]
=> 2
[[[],[],[]]]
=> 3
[[[],[[]]]]
=> 2
[[[[]],[]]]
=> 2
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 5
[[],[],[],[[]]]
=> 4
[[],[],[[]],[]]
=> 4
[[],[],[[],[]]]
=> 3
[[],[],[[[]]]]
=> 3
[[],[[]],[],[]]
=> 4
[[],[[]],[[]]]
=> 3
[[],[[],[]],[]]
=> 3
[[],[[[]]],[]]
=> 3
[[],[[],[],[]]]
=> 3
[[],[[],[[]]]]
=> 2
[[],[[[]],[]]]
=> 2
[[],[[[],[]]]]
=> 2
[[],[[[[]]]]]
=> 2
[[[]],[],[],[]]
=> 4
[[[]],[],[[]]]
=> 3
[[[]],[[]],[]]
=> 3
[[[]],[[],[]]]
=> 2
[[[]],[[[]]]]
=> 2
[[[],[]],[],[]]
=> 3
[[[[]]],[],[]]
=> 3
[[[],[]],[[]]]
=> 2
[[[[]]],[[]]]
=> 2
[[[],[],[]],[]]
=> 3
[[[],[[]]],[]]
=> 2
[[[[]],[]],[]]
=> 2
[[[[],[]]],[]]
=> 2
[[[[[]]]],[]]
=> 2
[[[]],[],[],[],[],[]]
=> ? = 6
[[[]],[],[],[],[[]]]
=> ? = 5
[[[]],[],[],[[]],[]]
=> ? = 5
[[[]],[],[],[[],[]]]
=> ? = 4
[[[]],[],[],[[[]]]]
=> ? = 4
[[[]],[],[[]],[],[]]
=> ? = 5
[[[]],[],[[]],[[]]]
=> ? = 4
[[[]],[],[[],[]],[]]
=> ? = 4
[[[]],[],[[],[],[]]]
=> ? = 3
[[[]],[],[[],[[]]]]
=> ? = 3
[[[]],[[]],[],[],[]]
=> ? = 5
[[[]],[[],[],[],[]]]
=> ? = 4
[[[],[]],[],[],[],[]]
=> ? = 5
[[[],[],[],[],[]],[]]
=> ? = 5
[[[],[],[],[],[],[]]]
=> ? = 6
[[[],[],[],[],[[]]]]
=> ? = 5
[[[],[],[],[[]],[]]]
=> ? = 5
[[[],[],[],[[],[]]]]
=> ? = 4
[[[],[],[],[[[]]]]]
=> ? = 4
[[[],[],[[]],[],[]]]
=> ? = 5
[[[],[],[[]],[[]]]]
=> ? = 4
[[[],[],[[],[]],[]]]
=> ? = 4
[[[],[],[[],[],[]]]]
=> ? = 3
[[[],[],[[],[[]]]]]
=> ? = 3
[[[],[[]],[],[],[]]]
=> ? = 5
[[[],[[],[],[],[]]]]
=> ? = 4
[[[]],[],[],[],[],[],[]]
=> ? = 7
[[[]],[],[],[],[],[[]]]
=> ? = 6
[[[]],[],[],[],[[]],[]]
=> ? = 6
[[[]],[],[],[],[[],[]]]
=> ? = 5
[[[]],[],[],[],[[[]]]]
=> ? = 5
[[[]],[],[],[[]],[],[]]
=> ? = 6
[[[]],[],[],[[],[],[]]]
=> ? = 4
[[[],[],[],[],[],[],[]]]
=> ? = 7
[[[],[],[],[],[],[[]]]]
=> ? = 6
[[[],[],[],[],[[]],[]]]
=> ? = 6
[[[],[],[],[],[[],[]]]]
=> ? = 5
[[[],[],[],[],[[[]]]]]
=> ? = 5
[[[],[],[],[[]],[],[]]]
=> ? = 6
[[[],[],[],[[],[],[]]]]
=> ? = 4
Description
The maximum number of child nodes in a tree.
Matching statistic: St000846
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 86%
St000846: Posets ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 86%
Values
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[[[]],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 6
[[[]],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 5
[[[]],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 5
[[[]],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4
[[[]],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[[]],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 5
[[[]],[],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 4
[[[]],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4
[[[]],[],[[],[],[]]]
=> ([(0,7),(1,6),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3
[[[]],[],[[],[[]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,7),(5,6),(6,7)],8)
=> ? = 3
[[[]],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 5
[[[]],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 4
[[[],[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 5
[[[],[],[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 5
[[[],[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ? = 6
[[[],[],[],[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ? = 5
[[[],[],[],[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ? = 5
[[[],[],[],[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,7),(7,5)],8)
=> ? = 4
[[[],[],[],[[[]]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,6),(6,7),(7,5)],8)
=> ? = 4
[[[],[],[[]],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ? = 5
[[[],[],[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,7),(7,6)],8)
=> ? = 4
[[[],[],[[],[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,7),(7,5)],8)
=> ? = 4
[[[],[],[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(6,7),(7,5)],8)
=> ? = 3
[[[],[],[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,5),(5,6),(6,7),(7,4)],8)
=> ? = 3
[[[],[[]],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ? = 5
[[[],[[],[],[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(6,5),(7,6)],8)
=> ? = 4
[[[]],[],[],[],[],[],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 7
[[[]],[],[],[],[],[[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 6
[[[]],[],[],[],[[]],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 6
[[[]],[],[],[],[[],[]]]
=> ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 5
[[[]],[],[],[],[[[]]]]
=> ([(0,8),(1,8),(2,8),(3,5),(4,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[[[]],[],[],[[]],[],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 6
[[[]],[],[],[[],[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 4
[[[],[],[],[],[],[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(8,7)],9)
=> ? = 7
[[[],[],[],[],[],[[]]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,8),(8,7)],9)
=> ? = 6
[[[],[],[],[],[[]],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,8),(8,7)],9)
=> ? = 6
[[[],[],[],[],[[],[]]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(7,8),(8,6)],9)
=> ? = 5
[[[],[],[],[],[[[]]]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(7,8),(8,6)],9)
=> ? = 5
[[[],[],[],[[]],[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,8),(8,7)],9)
=> ? = 6
[[[],[],[],[[],[],[]]]]
=> ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(7,8),(8,6)],9)
=> ? = 4
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000845
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 5
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> 3
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> 3
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> 3
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 3
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> 3
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> 3
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> 3
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 3
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> 2
[[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[[[]],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(7,1)],8)
=> ? = 6
[[[]],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(6,2),(7,1)],8)
=> ? = 5
[[[]],[],[],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(6,2),(7,1)],8)
=> ? = 5
[[[]],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 4
[[[]],[],[],[[[]]]]
=> ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,6),(0,7),(5,1),(6,2),(7,5)],8)
=> ? = 4
[[[]],[],[[]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(6,2),(7,1)],8)
=> ? = 5
[[[]],[],[[]],[[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4
[[[]],[],[[],[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 4
[[[]],[],[[],[],[]]]
=> ([(0,7),(1,6),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 3
[[[]],[],[[],[[]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 3
[[[]],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(6,2),(7,1)],8)
=> ? = 5
[[[]],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(7,6)],8)
=> ?
=> ? = 4
[[[],[]],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(7,1),(7,2)],8)
=> ? = 5
[[[],[],[],[],[]],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(7,1),(7,2),(7,3),(7,4),(7,5)],8)
=> ? = 5
[[[],[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6)],8)
=> ? = 6
[[[],[],[],[],[[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,7),(6,5),(7,1),(7,2),(7,3),(7,4),(7,6)],8)
=> ? = 5
[[[],[],[],[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,7),(6,5),(7,1),(7,2),(7,3),(7,4),(7,6)],8)
=> ? = 5
[[[],[],[],[[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,7),(7,5)],8)
=> ([(0,7),(6,4),(6,5),(7,1),(7,2),(7,3),(7,6)],8)
=> ? = 4
[[[],[],[],[[[]]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,6),(6,7),(7,5)],8)
=> ([(0,7),(5,6),(6,4),(7,1),(7,2),(7,3),(7,5)],8)
=> ? = 4
[[[],[],[[]],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,7),(6,5),(7,1),(7,2),(7,3),(7,4),(7,6)],8)
=> ? = 5
[[[],[],[[]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,7),(7,6)],8)
=> ([(0,7),(5,4),(6,3),(7,1),(7,2),(7,5),(7,6)],8)
=> ? = 4
[[[],[],[[],[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,7),(7,5)],8)
=> ([(0,7),(6,4),(6,5),(7,1),(7,2),(7,3),(7,6)],8)
=> ? = 4
[[[],[],[[],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,6),(6,7),(7,5)],8)
=> ([(0,7),(6,3),(6,4),(6,5),(7,1),(7,2),(7,6)],8)
=> ? = 3
[[[],[],[[],[[]]]]]
=> ([(0,7),(1,7),(2,6),(3,5),(5,6),(6,7),(7,4)],8)
=> ([(0,7),(5,4),(6,3),(6,5),(7,1),(7,2),(7,6)],8)
=> ? = 3
[[[],[[]],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,7),(6,5),(7,1),(7,2),(7,3),(7,4),(7,6)],8)
=> ? = 5
[[[],[[],[],[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(6,5),(7,6)],8)
=> ([(0,6),(6,5),(6,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 4
[[[]],[],[],[],[],[],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(8,1)],9)
=> ? = 7
[[[]],[],[],[],[],[[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? = 6
[[[]],[],[],[],[[]],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? = 6
[[[]],[],[],[],[[],[]]]
=> ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? = 5
[[[]],[],[],[],[[[]]]]
=> ([(0,8),(1,8),(2,8),(3,5),(4,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? = 5
[[[]],[],[],[[]],[],[]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? = 6
[[[]],[],[],[[],[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? = 4
[[[],[],[],[],[],[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(8,7)],9)
=> ([(0,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6),(8,7)],9)
=> ? = 7
[[[],[],[],[],[],[[]]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(7,6),(8,1),(8,2),(8,3),(8,4),(8,5),(8,7)],9)
=> ? = 6
[[[],[],[],[],[[]],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(7,6),(8,1),(8,2),(8,3),(8,4),(8,5),(8,7)],9)
=> ? = 6
[[[],[],[],[],[[],[]]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(7,8),(8,6)],9)
=> ([(0,8),(7,5),(7,6),(8,1),(8,2),(8,3),(8,4),(8,7)],9)
=> ? = 5
[[[],[],[],[],[[[]]]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,5),(5,7),(7,8),(8,6)],9)
=> ([(0,8),(6,7),(7,5),(8,1),(8,2),(8,3),(8,4),(8,6)],9)
=> ? = 5
[[[],[],[],[[]],[],[]]]
=> ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(7,6),(8,1),(8,2),(8,3),(8,4),(8,5),(8,7)],9)
=> ? = 6
[[[],[],[],[[],[],[]]]]
=> ([(0,8),(1,8),(2,8),(3,7),(4,7),(5,7),(7,8),(8,6)],9)
=> ?
=> ? = 4
Description
The maximal number of elements covered by an element in a poset.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
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