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Your data matches 54 different statistics following compositions of up to 3 maps.
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Matching statistic: St000390
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00088: Permutations āKreweras complementā¶ Permutations
Mp00114: Permutations āconnectivity setā¶ Binary words
St000390: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00088: Permutations āKreweras complementā¶ Permutations
Mp00114: Permutations āconnectivity setā¶ Binary words
St000390: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => 1 => 1
[1,0,1,0]
=> [3,1,2] => [3,1,2] => 00 => 0
[1,1,0,0]
=> [2,3,1] => [1,2,3] => 11 => 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 000 => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => 001 => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 000 => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,1,3,2] => 000 => 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 111 => 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 0000 => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,4,1,2,5] => 0001 => 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => 0000 => 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,5,1,4,2] => 0000 => 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => 0000 => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => 0001 => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,5,3,1,2] => 0000 => 0
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,1,3,2] => 0000 => 0
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,1,3,2,5] => 0001 => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 0000 => 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,2,1,4,3] => 0000 => 0
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => 0000 => 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [3,4,5,1,2,6] => 00001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,4,6,2,1,5] => 00000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [3,4,6,1,5,2] => 00000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,4,1,2,5,6] => 00011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,5,2,6,1,4] => 00000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,5,2,1,4,6] => 00001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [3,5,6,4,1,2] => 00000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [3,5,6,1,4,2] => 00000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [3,5,1,4,2,6] => 00001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,6,2,4,1,5] => 00000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,6,2,1,5,4] => 00000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,6,1,4,5,2] => 00000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,2,4,5,6] => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [4,2,5,6,1,3] => 00000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,2,5,1,3,6] => 00001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,6,3,1,5] => 00000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,2,6,1,5,3] => 00000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,1,3,5,6] => 00011 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [4,5,3,6,1,2] => 00000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,5,3,1,2,6] => 00001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,6,3,1,2] => 00000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [4,5,6,1,2,3] => 00000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,5,1,3,2,6] => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,6,3,2,1,5] => 00000 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,6,3,1,5,2] => 00000 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,1,3,5,2] => 00000 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,1,3,2,5,6] => 00011 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000297
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00028: Dyck paths āreverseā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 1001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,2,6,7,8,9,10] => ? => ? = 0
Description
The number of leading ones in a binary word.
Matching statistic: St000382
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00064: Permutations āreverseā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 94% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 100%
Mp00064: Permutations āreverseā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 94% āvalues known / values provided: 94%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [2] => 2 = 1 + 1
[1,0,1,0]
=> [3,1,2] => [2,1,3] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,3,1] => [1,3,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [1,2,1] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [2,1,3,4] => [1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => [2,1,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [4,3,2,1,5] => [1,1,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,2,1,4] => [2,1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,2,4,1,5] => [1,2,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,1,3,5] => [1,1,3] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2,1,4,5] => [1,1,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,5,1,3,4] => [2,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,2,1,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,1,4,5,2] => [1,3,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => [1,2,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [2,1,1,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => [1,1,1,1,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [4,6,3,2,1,5] => [2,1,1,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,3,6,2,1,4] => [1,2,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [4,3,5,2,1,6] => [1,2,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => [2,1,1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [5,4,2,6,1,3] => [1,1,2,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => [2,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [5,3,2,4,1,6] => [1,1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,3,2,5,1,6] => [1,1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [3,6,2,4,1,5] => [2,2,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,2,6,4,1,3] => [1,2,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,2,5,6,1,3] => [1,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,2,5,4,1,6] => [1,2,1,2] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,5,4,1,3] => [2,1,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,4,3,1,6,2] => [1,1,1,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,6,3,1,5,2] => [2,1,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => [1,2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,3,5,1,6,2] => [1,2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,6,5,1,4,2] => [2,1,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [5,4,2,1,3,6] => [1,1,1,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,6,2,1,3,5] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [5,3,2,1,4,6] => [1,1,1,3] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => [1,1,1,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,6,2,1,4,5] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,6,1,3,4] => [1,2,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,2,5,1,3,6] => [1,2,3] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,2,5,1,4,6] => [1,2,3] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [2,6,5,1,3,4] => [2,1,3] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [3,1,8,5,6,2,4,7] => [7,4,2,6,5,8,1,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [8,1,4,5,6,2,3,7] => [7,3,2,6,5,4,1,8] => ? => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,1,7,6,3,8,5] => [5,8,3,6,7,1,4,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,8,1,5,6,3,4,7] => [7,4,3,6,5,1,8,2] => ? => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [7,3,1,2,6,4,8,5] => [5,8,4,6,2,1,3,7] => ? => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [2,3,8,6,1,4,5,7] => [7,5,4,1,6,8,3,2] => ? => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [9,1,2,3,4,7,5,6,8] => [8,6,5,7,4,3,2,1,9] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,9,2,3,4,5,6] => [6,5,4,3,2,9,8,1,7] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,5,1,3,6,7,8,9,4] => [4,9,8,7,6,3,1,5,2] => ? => ? = 1 + 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,6,1,2,3,7,8,9,4] => [4,9,8,7,3,2,1,6,5] => ? => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [6,3,4,1,2,7,8,9,5] => [5,9,8,7,2,1,4,3,6] => ? => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [9,3,7,1,2,4,5,6,8] => [8,6,5,4,2,1,7,3,9] => ? => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,7,9,1,2,3,4,5,8] => [8,5,4,3,2,1,9,7,6] => ? => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,7,8,1,2,3,4,9,5] => [5,9,4,3,2,1,8,7,6] => ? => ? = 1 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,7,8,9,1,3,4,5,6] => [6,5,4,3,1,9,8,7,2] => ? => ? = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,1,7,8,2,6] => [6,2,8,7,1,5,4,3,9] => ? => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [8,9,4,5,6,1,2,3,7] => [7,3,2,1,6,5,4,9,8] => ? => ? = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,1,8,2,7] => [7,2,8,1,6,5,4,3,9] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [10,1,11,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1,1] => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [2,10,4,5,6,7,8,1,3,9] => [9,3,1,8,7,6,5,4,10,2] => [1,1,2,1,1,1,2,1] => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,10,5,6,7,8,9,2,4] => [4,2,9,8,7,6,5,10,1,3] => [1,2,1,1,1,2,2] => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,10,3] => [3,10,2,8,7,6,5,4,1,9] => [2,2,1,1,1,1,2] => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,7,8,11,1,9,10] => [10,9,1,11,8,7,6,5,4,3,2] => ? => ? = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
=> [10,3,4,5,6,7,1,9,2,8] => [8,2,9,1,7,6,5,4,3,10] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => [2,1,9,8,7,6,5,4,3,10] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,9,3,4,5,6,7,8] => [8,7,6,5,4,3,9,2,1,10] => [1,1,1,1,1,2,1,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [9,11,8,7,6,5,4,3,2,1,10] => [2,1,1,1,1,1,1,1,2] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,11,6,7,8,9] => [9,8,7,6,11,5,4,3,2,1,10] => [1,1,1,2,1,1,1,1,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,10,5,6,7,8,9] => [9,8,7,6,5,10,4,3,2,1,11] => [1,1,1,1,2,1,1,1,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,10,9,3,5,6,7,8] => [8,7,6,5,3,9,10,2,1,4] => [1,1,1,1,3,1,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,10,4,5,6,7,8,9] => [9,8,7,6,5,4,10,3,2,1,11] => [1,1,1,1,1,2,1,1,2] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,10,9,2,4,5,6,7,8] => [8,7,6,5,4,2,9,10,1,3] => [1,1,1,1,1,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,11,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,11,2,1,10] => [1,1,1,1,1,1,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1,11] => [1,1,1,1,1,1,1,2,2] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,10,1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1,10,11] => [1,1,1,1,1,1,1,1,3] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [11,9,1,2,3,4,5,6,7,8,10] => [10,8,7,6,5,4,3,2,1,9,11] => [1,1,1,1,1,1,1,1,3] => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [10,7,8,1,2,3,4,5,6,9] => [9,6,5,4,3,2,1,8,7,10] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [11,3,4,5,6,7,8,9,10,1,2] => [2,1,10,9,8,7,6,5,4,3,11] => ? => ? = 0 + 1
[1,1,1,0,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [6,3,4,1,2,7,8,9,10,5] => [5,10,9,8,7,2,1,4,3,6] => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [11,1,2,3,4,5,9,6,7,8,10] => [10,8,7,6,9,5,4,3,2,1,11] => ? => ? = 0 + 1
Description
The first part of an integer composition.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00064: Permutations āreverseā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 91% āvalues known / values provided: 91%ādistinct values known / distinct values provided: 100%
Mp00064: Permutations āreverseā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 91% āvalues known / values provided: 91%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => 0 => 2 = 1 + 1
[1,0,1,0]
=> [3,1,2] => [2,1,3] => 10 => 1 = 0 + 1
[1,1,0,0]
=> [2,3,1] => [1,3,2] => 01 => 2 = 1 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 110 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 010 => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 101 => 1 = 0 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [2,1,3,4] => 100 => 1 = 0 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [4,3,2,1,5] => 1110 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,2,1,4] => 0110 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 1010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,2,4,1,5] => 1010 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 0110 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 1101 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 0101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1100 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2,1,4,5] => 1100 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,5,1,3,4] => 0100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 1011 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,1,4,5,2] => 1001 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1010 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 0111 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => 11110 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [4,6,3,2,1,5] => 01110 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,3,6,2,1,4] => 10110 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [4,3,5,2,1,6] => 10110 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 01110 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [5,4,2,6,1,3] => 11010 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 01010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [5,3,2,4,1,6] => 11010 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,3,2,5,1,6] => 11010 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [3,6,2,4,1,5] => 01010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,2,6,4,1,3] => 10110 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,2,5,6,1,3] => 10010 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,2,5,4,1,6] => 10110 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,5,4,1,3] => 01110 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,4,3,1,6,2] => 11101 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,6,3,1,5,2] => 01101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 10101 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,3,5,1,6,2] => 10101 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,6,5,1,4,2] => 01101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [5,4,2,1,3,6] => 11100 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,6,2,1,3,5] => 01100 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [5,3,2,1,4,6] => 11100 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => 11101 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,6,2,1,4,5] => 01100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,6,1,3,4] => 10100 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,2,5,1,3,6] => 10100 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,2,5,1,4,6] => 10100 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [2,6,5,1,3,4] => 01100 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [7,6,5,3,8,2,1,4] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,8,2,4,7,5,6] => [6,5,7,4,2,8,1,3] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,1,8,2,7,4,5,6] => [6,5,4,7,2,8,1,3] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => [7,4,8,6,2,5,1,3] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => [4,8,7,6,2,5,1,3] => ? => ? = 1 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [8,1,5,2,6,7,3,4] => [4,3,7,6,2,5,1,8] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,7,5,2,4,8,6] => [6,8,4,2,5,7,1,3] => ? => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [8,1,4,6,2,3,5,7] => [7,5,3,2,6,4,1,8] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => [7,4,3,2,5,6,1,8] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => [7,6,2,8,5,4,1,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [3,1,8,5,6,2,4,7] => [7,4,2,6,5,8,1,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [8,1,4,5,6,2,3,7] => [7,3,2,6,5,4,1,8] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,8,1,3,6,4,5,7] => [7,5,4,6,3,1,8,2] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,7,1,3,6,4,8,5] => [5,8,4,6,3,1,7,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => [7,6,5,3,8,1,4,2] => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => [6,8,5,3,7,1,4,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,8,1,5,3,4,6,7] => [7,6,4,3,5,1,8,2] => ? => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,1,7,6,3,8,5] => [5,8,3,6,7,1,4,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,8,1,5,6,3,4,7] => [7,4,3,6,5,1,8,2] => ? => ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,8,7,3,6] => [6,3,7,8,5,1,4,2] => ? => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [5,3,1,2,8,4,6,7] => [7,6,4,8,2,1,3,5] => ? => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [7,3,1,2,6,4,8,5] => [5,8,4,6,2,1,3,7] => ? => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,1,2,3,8,5,7] => [7,5,8,3,2,1,4,6] => ? => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [8,3,1,5,2,4,6,7] => [7,6,4,2,5,1,3,8] => ? => ? = 0 + 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,1,5,2,8,4,7] => [7,4,8,2,5,1,3,6] => ? => ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,1,5,7,2,8,6] => [6,8,2,7,5,1,3,4] => ? => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [2,8,4,1,3,7,5,6] => [6,5,7,3,1,4,8,2] => ? => ? = 0 + 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,8,5,1,6,3,4,7] => [7,4,3,6,1,5,8,2] => ? => ? = 0 + 1
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [7,3,4,1,2,5,8,6] => [6,8,5,2,1,4,3,7] => ? => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,8,4,1,2,3,5,7] => [7,5,3,2,1,4,8,6] => ? => ? = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [8,3,4,1,6,2,5,7] => [7,5,2,6,1,4,3,8] => ? => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [2,3,8,6,1,4,5,7] => [7,5,4,1,6,8,3,2] => ? => ? = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [8,3,4,6,1,2,5,7] => [7,5,2,1,6,4,3,8] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,9,2,3,4,5,6] => [6,5,4,3,2,9,8,1,7] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,5,1,3,6,7,8,9,4] => [4,9,8,7,6,3,1,5,2] => ? => ? = 1 + 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,6,1,2,3,7,8,9,4] => [4,9,8,7,3,2,1,6,5] => ? => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [6,3,4,1,2,7,8,9,5] => [5,9,8,7,2,1,4,3,6] => ? => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [9,3,7,1,2,4,5,6,8] => [8,6,5,4,2,1,7,3,9] => ? => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,7,9,1,2,3,4,5,8] => [8,5,4,3,2,1,9,7,6] => ? => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,7,8,1,2,3,4,9,5] => [5,9,4,3,2,1,8,7,6] => ? => ? = 1 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,7,8,9,1,3,4,5,6] => [6,5,4,3,1,9,8,7,2] => ? => ? = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,1,7,8,2,6] => [6,2,8,7,1,5,4,3,9] => ? => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [8,9,4,5,6,1,2,3,7] => [7,3,2,1,6,5,4,9,8] => ? => ? = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [9,3,4,5,6,1,8,2,7] => [7,2,8,1,6,5,4,3,9] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [10,1,11,9,8,7,6,5,4,3,2] => 1011111111 => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,7,8,11,1,9,10] => [10,9,1,11,8,7,6,5,4,3,2] => ? => ? = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
=> [10,3,4,5,6,7,1,9,2,8] => [8,2,9,1,7,6,5,4,3,10] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => [2,1,9,8,7,6,5,4,3,10] => ? => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [9,11,8,7,6,5,4,3,2,1,10] => 0111111110 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [11,1,2,3,4,10,5,6,7,8,9] => [9,8,7,6,5,10,4,3,2,1,11] => 1111011110 => ? = 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: St000745
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00028: Dyck paths āreverseā¶ Dyck paths
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 88% āvalues known / values provided: 88%ādistinct values known / distinct values provided: 100%
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 88% āvalues known / values provided: 88%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => [[1,2],[3]]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[1,2,4],[3],[5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [[1,2,3],[4],[5]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [[1,2],[3,5],[4]]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [[1,2,4,5],[3,6]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [[1,2,3,5],[4,6]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [[1,2,3,5],[4],[6]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [[1,3,5],[2,4],[6]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [[1,2,5],[3,4,6]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [[1,2,5],[3,6],[4]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [[1,2,5],[3,4],[6]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [[1,2,3,4],[5,6]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [[1,2,4],[3,5,6]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [[1,2,4],[3,6],[5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [[1,3,4,6],[2,5]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [[1,2,3,4],[5],[6]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [[1,3,4],[2,5],[6]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [[1,2,3,4],[5],[6]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [[1,3,4],[2,5],[6]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [[1,2,4],[3,5],[6]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [[1,2,4],[3,5],[6]]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [[1,2,4],[3,5],[6]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [[1,3,4],[2,5],[6]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [9,1,4,2,3,5,6,7,8] => ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,3,1,9,2,5,6,7,8] => [[1,2,5,6,7,8],[3,9],[4]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => [[1,2,5,6,7,8],[3,4],[9]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,1,2,9,3,6,7,8] => [[1,2,3,6,7,8],[4,9],[5]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [9,4,1,5,2,3,6,7,8] => [[1,2,3,6,7,8],[4,5],[9]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,1,5,9,2,6,7,8] => [[1,2,6,7,8],[3,5,9],[4]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [9,3,4,5,1,2,6,7,8] => ?
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [5,6,1,2,3,9,4,7,8] => [[1,2,3,4,7,8],[5,6,9]]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4,1,2,6,9,3,7,8] => [[1,2,3,7,8],[4,6,9],[5]]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,6,1,2,3,4,9,5,8] => [[1,2,3,4,5,8],[6,9],[7]]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [9,6,1,2,3,7,4,5,8] => [[1,2,3,4,5,8],[6,7],[9]]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,7,9,1,2,3,4,5,8] => [[1,2,3,4,5,8],[6,7,9]]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,9,4,8] => [[1,2,3,4,8],[5,6,7,9]]
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [2,3,9,5,6,7,1,4,8] => [[1,3,4,6,7,8],[2,5],[9]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [9,7,4,5,6,1,2,3,8] => [[1,2,3,8],[4,5,6],[7],[9]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [[1,3,5,6,7,8],[2,4],[9]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [[1,2,5,7,8],[3,4,9],[6]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [[1,2,5,6,8],[3,4,9],[7]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [[1,2,5,6,7,8],[3,4],[9]]
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,5,8,1,6,9,7] => [[1,3,4,5,6,7],[2,8,9]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [2,3,4,8,9,1,5,6,7] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [8,7,1,2,3,4,5,9,6] => [[1,2,3,4,5,6],[7,9],[8]]
=> ? = 0 + 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [2,3,4,9,1,7,8,5,6] => [[1,3,4,5,6],[2,7,8],[9]]
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [9,1,7,2,3,4,8,5,6] => ?
=> ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,7,8,9,1,3,4,5,6] => [[1,3,4,5,6],[2,7,8,9]]
=> ? = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [7,6,1,2,3,4,8,9,5] => [[1,2,3,4,5,9],[6,8],[7]]
=> ? = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [[1,2,5,7,8],[3,4,6],[9]]
=> ? = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,1,6,7,8,2,5] => ?
=> ? = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [[1,2,4,7,8],[3,5,6],[9]]
=> ? = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [9,3,1,5,6,7,8,2,4] => [[1,2,4,7,8],[3,5,6],[9]]
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,1,4,5,6,2,8,9,3] => [[1,2,3,6,8,9],[4,5],[7]]
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [[1,2,3,6,7,9],[4,5],[8]]
=> ? = 0 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [[1,2,3,6,7,8],[4,5],[9]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => [[1,2,5,6,7,8],[3,4],[9]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => [[1,2,3,6,7,8,9],[4,5],[10]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [[1,2,5,6,7,8,9],[3,4],[10]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [11,1,4,5,6,7,8,9,10,2,3] => [[1,2,3,6,7,8,9,10],[4,5],[11]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,10,3] => [[1,2,3,6,7,8,10],[4,5],[9]]
=> ? = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,10,5,6,7,8,9,2,4] => [[1,2,4,7,8,9],[3,5,6],[10]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,3,4,5,6,7,1,10,2,9] => [[1,2,5,6,7,9],[3,4,10],[8]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [2,10,4,5,6,7,8,1,3,9] => [[1,3,5,6,7,8,9],[2,4],[10]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,4,5,6,7,8,9,1,2,10] => [[1,2,5,6,7,8,9,10],[3,4],[11]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,10,3] => [[1,2,3,6,7,8,9,10],[4,5]]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,7,10,1,8,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,10,11,3] => ?
=> ? = 0 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [10,3,1,5,6,7,8,9,2,4] => [[1,2,4,7,8,9],[3,5,6],[10]]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => [[1,2,5,6,7,8,9],[3,4],[10]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,10,1,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 1 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000678
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00072: Permutations ābinary search tree: left to rightā¶ Binary trees
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
St000678: Dyck paths ā¶ ā¤Result quality: 84% āvalues known / values provided: 84%ādistinct values known / distinct values provided: 100%
Mp00072: Permutations ābinary search tree: left to rightā¶ Binary trees
Mp00020: Binary trees āto Tamari-corresponding Dyck pathā¶ Dyck paths
St000678: Dyck paths ā¶ ā¤Result quality: 84% āvalues known / values provided: 84%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 1 + 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,3,6,7,8,4] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,3,7,8,4] => [.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,3,4,7,8,5] => [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,2,6,7,8,3] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,8,2,3] => [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,3,6,7,8,4] => [.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,2,3,4,6,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,2,3,8,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [4,1,5,6,7,8,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [4,7,1,2,3,8,5,6] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,1,2,3,4,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [5,6,7,1,2,3,8,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [5,7,1,2,3,4,8,6] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,2,7,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 0 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths āRingelā¶ Permutations
Mp00149: Permutations āLehmer code rotationā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 83% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 100%
Mp00149: Permutations āLehmer code rotationā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 83% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [2] => 2 = 1 + 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => [3,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [1,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [1,2,1] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,2,4,3] => [3,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => [2,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,3,4,5,2] => [4,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,1,5,3] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,3,2,5,4] => [2,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [4,2,5,1,3] => [1,2,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [1,3,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,2,1,4] => [2,1,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,3,4,2] => [2,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,4,5,3] => [4,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,4,2,1,3] => [1,1,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [3,4,1,5,2] => [2,2,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,1,2,5,4] => [1,3,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [2,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [3,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,3,4,5,6,2] => [5,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,2,3,4,1,5] => [1,3,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,2,3,1,6,4] => [1,2,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,3,4,2,6,5] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [5,2,3,6,1,4] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,2,1,5,6,3] => [1,1,3,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,2,6,3,1,5] => [1,2,1,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,3,6,4,5,2] => [3,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,3,2,5,6,4] => [2,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,2,5,3,1,4] => [1,2,1,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,2,5,1,6,3] => [1,2,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,2,1,3,6,5] => [1,1,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,3,6,2,5,4] => [3,2,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,2,5,6,1,3] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => [1,4,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,2,4,1,5] => [2,2,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,5,2,1,6,4] => [2,1,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,4,2,6,5] => [1,2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,5,2,6,1,4] => [2,2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,5,3,4,6,2] => [2,3,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => [1,1,2,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,3,4,5,2] => [2,3,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,3,4,5,2] => [1,4,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [6,5,2,3,1,4] => [1,1,2,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,4,2,1,6,3] => [1,1,1,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,5,3,2,6,4] => [2,1,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,5,4] => [2,1,2,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [5,4,2,6,1,3] => [1,1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => [1,3,2,5,6,7,8,4] => [2,5,1] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [8,1,5,2,3,7,4,6] => [1,3,7,4,5,2,8,6] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [8,1,4,2,7,3,5,6] => [1,3,6,4,2,7,8,5] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => [1,3,8,7,4,2,6,5] => [3,1,1,2,1] => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [8,1,4,5,6,2,3,7] => [1,3,6,7,8,4,5,2] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,6,1,3,4,8,5,7] => [3,7,2,4,5,1,8,6] => ? => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,8,3,7,5,6] => [3,5,2,1,6,4,8,7] => ? => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,8,1,5,3,4,6,7] => [3,1,4,7,5,6,8,2] => ? => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,8,1,5,6,3,4,7] => [3,1,4,7,8,5,6,2] => ? => ? = 0 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,7,1,5,6,3,8,4] => [3,8,2,6,7,4,1,5] => ? => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,5,3,4,6,7,8,2] => [2,5,1] => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,6,3,4,5,7,8,2] => [2,5,1] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => [1,8,3,4,5,6,7,2] => [2,5,1] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => [8,7,2,3,4,5,1,6] => [1,1,4,2] => ? = 1 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => [6,5,2,3,7,8,1,4] => [1,1,4,2] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [8,3,1,5,2,4,6,7] => [1,5,3,7,4,6,8,2] => ? => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => [8,4,2,1,5,6,7,3] => [1,1,1,4,1] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [7,4,1,8,2,3,5,6] => [8,5,2,1,4,6,7,3] => [1,1,1,4,1] => ? = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,1,8,7,2,5,6] => [5,4,2,1,3,7,8,6] => [1,1,1,4,1] => ? = 0 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => [5,4,2,6,7,8,1,3] => [1,1,4,2] => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => [3,4,1,5,6,7,8,2] => [2,5,1] => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [2,6,8,1,3,4,5,7] => [3,7,1,4,5,6,8,2] => [2,5,1] => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,3,8,1,2,4,5,7] => [7,4,1,3,5,6,8,2] => ? => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => [1,5,2,4,6,7,8,3] => [2,5,1] => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => [7,8,1,3,4,5,6,2] => [2,5,1] => ? = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,5,4,1,2,8,3,7] => [7,6,5,2,3,1,8,4] => ? => ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [5,3,4,1,8,2,6,7] => [6,4,5,2,1,7,8,3] => ? => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [2,8,6,5,1,3,4,7] => [3,1,8,7,4,5,6,2] => ? => ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,7,8,6,1,3,4,5] => [3,8,1,2,5,6,7,4] => [2,5,1] => ? = 0 + 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [8,3,6,5,1,7,2,4] => [1,5,8,7,3,2,6,4] => [3,1,1,2,1] => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => [3,4,5,6,8,2,1,7] => [5,1,2] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,3,4,7,6,1,8,5] => [3,4,5,8,7,2,1,6] => [4,1,1,2] => ? = 1 + 1
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,3,8,7,6,1,4,5] => [3,4,1,2,5,7,8,6] => [2,5,1] => ? = 0 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [3,4,5,6,7,8,1,2] => [6,2] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [9,1,2,3,4,7,5,6,8] => [1,3,4,5,6,9,7,8,2] => ? => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => [1,3,4,5,6,2,8,9,7] => [5,3,1] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [9,1,2,3,6,8,4,5,7] => [1,3,4,5,8,2,7,9,6] => [5,3,1] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [9,1,2,3,6,7,8,4,5] => ? => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => [1,3,4,2,6,7,8,9,5] => [3,5,1] => ? = 0 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,5,6,7,8,9,3] => [5,2,3,6,7,8,9,1,4] => ? => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => [1,3,2,5,6,7,8,9,4] => [2,6,1] => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => [1,3,6,2,5,7,8,9,4] => [3,5,1] => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,9,2,3,4,5,6] => [8,2,9,1,4,5,6,7,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [9,1,8,5,6,7,2,3,4] => [1,3,2,8,9,4,6,7,5] => [2,3,3,1] => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => [4,2,5,1,8,9,3,7,6] => [1,2,3,2,1] => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [4,2,1,7,8,9,3,6,5] => [1,1,4,2,1] => ? = 0 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,5,1,3,6,7,8,9,4] => [3,6,2,4,7,8,9,1,5] => ? => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => [3,5,2,7,4,9,6,1,8] => [2,2,2,1,2] => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,5,3,4,6,7,8,9,2] => [2,6,1] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,7,1,2,3,4,5,6,8] => [1,9,3,4,5,6,7,8,2] => [2,6,1] => ? = 0 + 1
Description
The last part of an integer composition.
Matching statistic: St000986
Mp00102: Dyck paths ārise compositionā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00247: Graphs āde-duplicateā¶ Graphs
St000986: Graphs ā¶ ā¤Result quality: 83% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 100%
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00247: Graphs āde-duplicateā¶ Graphs
St000986: Graphs ā¶ ā¤Result quality: 83% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,1,0,0]
=> [2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ?
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,1,1,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001498
Mp00102: Dyck paths ārise compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St001498: Dyck paths ā¶ ā¤Result quality: 81% āvalues known / values provided: 81%ādistinct values known / distinct values provided: 100%
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St001498: Dyck paths ā¶ ā¤Result quality: 81% āvalues known / values provided: 81%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [3] => [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] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => [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] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000315
Mp00102: Dyck paths ārise compositionā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00247: Graphs āde-duplicateā¶ Graphs
St000315: Graphs ā¶ ā¤Result quality: 78% āvalues known / values provided: 78%ādistinct values known / distinct values provided: 100%
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00247: Graphs āde-duplicateā¶ Graphs
St000315: Graphs ā¶ ā¤Result quality: 78% āvalues known / values provided: 78%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,1,0,0]
=> [2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0
Description
The number of isolated vertices of a graph.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000989The number of final rises of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000237The number of small exceedances. St001271The competition number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St000654The first descent of a permutation. St000917The open packing number of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000234The number of global ascents of a permutation. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000546The number of global descents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000215The number of adjacencies of a permutation, zero appended.
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