searching the database
Your data matches 107 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000381
(load all 49 compositions to match this statistic)
(load all 49 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 2
{{1},{2}}
=> [1,1] => 1
{{1,2,3}}
=> [3] => 3
{{1,2},{3}}
=> [2,1] => 2
{{1,3},{2}}
=> [2,1] => 2
{{1},{2,3}}
=> [1,2] => 2
{{1},{2},{3}}
=> [1,1,1] => 1
{{1,2,3,4}}
=> [4] => 4
{{1,2,3},{4}}
=> [3,1] => 3
{{1,2,4},{3}}
=> [3,1] => 3
{{1,2},{3,4}}
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1] => 2
{{1,3,4},{2}}
=> [3,1] => 3
{{1,3},{2,4}}
=> [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1] => 2
{{1,4},{2,3}}
=> [2,2] => 2
{{1},{2,3,4}}
=> [1,3] => 3
{{1},{2,3},{4}}
=> [1,2,1] => 2
{{1,4},{2},{3}}
=> [2,1,1] => 2
{{1},{2,4},{3}}
=> [1,2,1] => 2
{{1},{2},{3,4}}
=> [1,1,2] => 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [5] => 5
{{1,2,3,4},{5}}
=> [4,1] => 4
{{1,2,3,5},{4}}
=> [4,1] => 4
{{1,2,3},{4,5}}
=> [3,2] => 3
{{1,2,3},{4},{5}}
=> [3,1,1] => 3
{{1,2,4,5},{3}}
=> [4,1] => 4
{{1,2,4},{3,5}}
=> [3,2] => 3
{{1,2,4},{3},{5}}
=> [3,1,1] => 3
{{1,2,5},{3,4}}
=> [3,2] => 3
{{1,2},{3,4,5}}
=> [2,3] => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => 2
{{1,2,5},{3},{4}}
=> [3,1,1] => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => 2
{{1,2},{3},{4,5}}
=> [2,1,2] => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 2
{{1,3,4,5},{2}}
=> [4,1] => 4
{{1,3,4},{2,5}}
=> [3,2] => 3
{{1,3,4},{2},{5}}
=> [3,1,1] => 3
{{1,3,5},{2,4}}
=> [3,2] => 3
{{1,3},{2,4,5}}
=> [2,3] => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => 2
{{1,3,5},{2},{4}}
=> [3,1,1] => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => 2
{{1,3},{2},{4,5}}
=> [2,1,2] => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 2
{{1,4,5},{2,3}}
=> [3,2] => 3
{{1,4},{2,3,5}}
=> [2,3] => 3
{{1,4},{2,3},{5}}
=> [2,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000444
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> 2
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 3
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> 1 = 2 - 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 0 = 1 - 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000983
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 10 => 01 => 2
{{1},{2}}
=> [1,1] => 11 => 11 => 1
{{1,2,3}}
=> [3] => 100 => 101 => 3
{{1,2},{3}}
=> [2,1] => 101 => 001 => 2
{{1,3},{2}}
=> [2,1] => 101 => 001 => 2
{{1},{2,3}}
=> [1,2] => 110 => 011 => 2
{{1},{2},{3}}
=> [1,1,1] => 111 => 111 => 1
{{1,2,3,4}}
=> [4] => 1000 => 0101 => 4
{{1,2,3},{4}}
=> [3,1] => 1001 => 1101 => 3
{{1,2,4},{3}}
=> [3,1] => 1001 => 1101 => 3
{{1,2},{3,4}}
=> [2,2] => 1010 => 1001 => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 0001 => 2
{{1,3,4},{2}}
=> [3,1] => 1001 => 1101 => 3
{{1,3},{2,4}}
=> [2,2] => 1010 => 1001 => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 0001 => 2
{{1,4},{2,3}}
=> [2,2] => 1010 => 1001 => 2
{{1},{2,3,4}}
=> [1,3] => 1100 => 1011 => 3
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 0011 => 2
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 0001 => 2
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 0011 => 2
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 0111 => 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 1111 => 1
{{1,2,3,4,5}}
=> [5] => 10000 => 10101 => 5
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 00101 => 4
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 00101 => 4
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 01101 => 3
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 11101 => 3
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 00101 => 4
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 01101 => 3
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 11101 => 3
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 01101 => 3
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 01001 => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 11001 => 2
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 11101 => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 11001 => 2
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 10001 => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 00001 => 2
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 00101 => 4
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 01101 => 3
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 11101 => 3
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 01101 => 3
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 01001 => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 11001 => 2
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 11101 => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 11001 => 2
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 10001 => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 00001 => 2
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 01101 => 3
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 01001 => 3
{{1,4},{2,3},{5}}
=> [2,2,1] => 10101 => 11001 => 2
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 10 => 01 => 1 = 2 - 1
{{1},{2}}
=> [1,1] => 11 => 00 => 0 = 1 - 1
{{1,2,3}}
=> [3] => 100 => 011 => 2 = 3 - 1
{{1,2},{3}}
=> [2,1] => 101 => 010 => 1 = 2 - 1
{{1,3},{2}}
=> [2,1] => 101 => 010 => 1 = 2 - 1
{{1},{2,3}}
=> [1,2] => 110 => 001 => 1 = 2 - 1
{{1},{2},{3}}
=> [1,1,1] => 111 => 000 => 0 = 1 - 1
{{1,2,3,4}}
=> [4] => 1000 => 0111 => 3 = 4 - 1
{{1,2,3},{4}}
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
{{1,2,4},{3}}
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
{{1,2},{3,4}}
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 0110 => 2 = 3 - 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
{{1},{2,3,4}}
=> [1,3] => 1100 => 0011 => 2 = 3 - 1
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 0001 => 1 = 2 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
{{1,2,3,4,5}}
=> [5] => 10000 => 01111 => 4 = 5 - 1
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 01110 => 3 = 4 - 1
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 01110 => 3 = 4 - 1
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 01100 => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 01110 => 3 = 4 - 1
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 01100 => 2 = 3 - 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 01100 => 2 = 3 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 01110 => 3 = 4 - 1
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 01100 => 2 = 3 - 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 01100 => 2 = 3 - 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 01101 => 2 = 3 - 1
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 01011 => 2 = 3 - 1
{{1,4},{2,3},{5}}
=> [2,2,1] => 10101 => 01010 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000521
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [[[]]]
=> 3 = 2 + 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [[],[]]
=> 2 = 1 + 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 4 = 3 + 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 3 = 2 + 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 3 = 2 + 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 3 = 2 + 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 5 = 4 + 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4 = 3 + 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4 = 3 + 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 2 + 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4 = 3 + 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 2 + 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 3 + 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 2 + 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 2 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 6 = 5 + 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5 = 4 + 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5 = 4 + 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4 = 3 + 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5 = 4 + 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4 = 3 + 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 3 = 2 + 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5 = 4 + 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4 = 3 + 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4 = 3 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 3 = 2 + 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 3 + 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
Description
The number of distinct subtrees of an ordered tree.
A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree $[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.
Matching statistic: St001039
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4},{5,6},{7,8}}
=> [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4},{5,7,8},{6}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1},{2},{3},{4},{5,6,7},{8}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1},{2},{3},{4},{5,8},{6,7}}
=> [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
{{1},{2},{3},{4},{5,6,8},{7}}
=> [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
{{1},{2},{3},{4,5},{6},{7,8}}
=> [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4,6,7,8},{5}}
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
{{1},{2},{3},{4,8},{5,6},{7}}
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1},{2},{3},{4,7,8},{5,6}}
=> [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
{{1},{2},{3},{4,8},{5,7},{6}}
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1},{2},{3},{4,8},{5,6,7}}
=> [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
{{1},{2},{3},{4,5,6,7,8}}
=> [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2},{3,4},{5},{6},{7,8}}
=> [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
{{1},{2},{3,4},{5},{6,8},{7}}
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
{{1},{2},{3,4},{5,6},{7},{8}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,4},{5,8},{6},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,4},{5,8},{6,7}}
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
{{1},{2},{3,5},{4},{6,7,8}}
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,8},{4},{5,6,7}}
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3
{{1},{2},{3,8},{4,5},{6},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,8},{4,6},{5},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,8},{4,7},{5},{6}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,8},{4,6,7},{5}}
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1},{2},{3,8},{4,5,6},{7}}
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1},{2},{3,8},{4,7},{5,6}}
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
{{1},{2},{3,5},{4,6},{7},{8}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3,5},{4,8},{6,7}}
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
{{1},{2},{3},{4},{5,7},{6,8}}
=> [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
{{1},{2},{3},{4,6},{5,8},{7}}
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1},{2},{3,5},{4,8},{6},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4,7},{5,8},{6}}
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1},{2},{3,6},{4,8},{5},{7}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4,6,7},{5,8}}
=> [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
{{1},{2},{3,7},{4,8},{5},{6}}
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1},{2},{3},{4,6,8},{5,7}}
=> [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000147
Mp00079: Set partitions —shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 2
{{1},{2}}
=> [1,1]
=> 1
{{1,2,3}}
=> [3]
=> 3
{{1,2},{3}}
=> [2,1]
=> 2
{{1,3},{2}}
=> [2,1]
=> 2
{{1},{2,3}}
=> [2,1]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> 2
{{1,3,4},{2}}
=> [3,1]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> 2
{{1,4},{2,3}}
=> [2,2]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> 2
{{1,4},{2},{3}}
=> [2,1,1]
=> 2
{{1},{2,4},{3}}
=> [2,1,1]
=> 2
{{1},{2},{3,4}}
=> [2,1,1]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> 3
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 3
{{1,2,4,5},{3}}
=> [4,1]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> 3
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 3
{{1,2,5},{3,4}}
=> [3,2]
=> 3
{{1,2},{3,4,5}}
=> [3,2]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 2
{{1,3,4,5},{2}}
=> [4,1]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> 3
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 3
{{1,3,5},{2,4}}
=> [3,2]
=> 3
{{1,3},{2,4,5}}
=> [3,2]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 2
{{1,4,5},{2,3}}
=> [3,2]
=> 3
{{1,4},{2,3,5}}
=> [3,2]
=> 3
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 2
{{1},{2},{3,4},{5,6},{7},{8}}
=> ?
=> ? = 2
{{1},{2},{3,4},{5,8},{6},{7}}
=> ?
=> ? = 2
{{1},{2},{3,4},{5,8},{6,7}}
=> ?
=> ? = 2
{{1},{2},{3,7,8},{4},{5},{6}}
=> ?
=> ? = 3
{{1},{2},{3,4,7,8},{5,6}}
=> ?
=> ? = 4
{{1},{2,3},{4},{5},{6,7},{8}}
=> ?
=> ? = 2
{{1},{2,3},{4},{5,8},{6,7}}
=> ?
=> ? = 2
{{1},{2,4},{3},{5,7},{6},{8}}
=> ?
=> ? = 2
{{1},{2,7,8},{3},{4},{5},{6}}
=> ?
=> ? = 3
{{1},{2,3,4},{5},{6,8},{7}}
=> ?
=> ? = 3
{{1},{2,3,4},{5,6,7},{8}}
=> ?
=> ? = 3
{{1},{2,7,8},{3,5,6},{4}}
=> ?
=> ? = 3
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ? = 6
{{1},{2,8},{3,4,7},{5},{6}}
=> ?
=> ? = 3
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ? = 6
{{1},{2,8},{3,4,5,6,7}}
=> ?
=> ? = 5
{{1,2},{3},{4},{5},{6,7},{8}}
=> ?
=> ? = 2
{{1,2},{3},{4},{5},{6,7,8}}
=> ?
=> ? = 3
{{1,2},{3},{4},{5,6},{7},{8}}
=> ?
=> ? = 2
{{1,2},{3},{4},{5,8},{6},{7}}
=> ?
=> ? = 2
{{1,2},{3},{4},{5,8},{6,7}}
=> ?
=> ? = 2
{{1,2},{3},{4,5},{6},{7},{8}}
=> ?
=> ? = 2
{{1,2},{3},{4,8},{5},{6},{7}}
=> ?
=> ? = 2
{{1,2},{3,4},{5},{6},{7},{8}}
=> ?
=> ? = 2
{{1,2},{3,5},{4},{6},{7},{8}}
=> ?
=> ? = 2
{{1,2},{3,6},{4},{5},{7},{8}}
=> ?
=> ? = 2
{{1,2},{3,8},{4},{5},{6},{7}}
=> ?
=> ? = 2
{{1,2},{3,4,5,7},{6},{8}}
=> ?
=> ? = 4
{{1,4},{2},{3},{5,8},{6,7}}
=> ?
=> ? = 2
{{1,5},{2},{3},{4},{6},{7,8}}
=> ?
=> ? = 2
{{1,8},{2},{3},{4},{5,6},{7}}
=> ?
=> ? = 2
{{1,5},{2},{3,4},{6,7,8}}
=> ?
=> ? = 3
{{1,8},{2},{3,4,5,6,7}}
=> ?
=> ? = 5
{{1,4},{2,3},{5},{6},{7,8}}
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,7},{8}}
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,8},{7}}
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,7,8}}
=> ?
=> ? = 3
{{1,4},{2,3},{5,6},{7},{8}}
=> ?
=> ? = 2
{{1,4},{2,3},{5,7},{6},{8}}
=> ?
=> ? = 2
{{1,4},{2,3},{5,8},{6},{7}}
=> ?
=> ? = 2
{{1,4},{2,3},{5,6,7},{8}}
=> ?
=> ? = 3
{{1,6},{2,3},{4},{5},{7},{8}}
=> ?
=> ? = 2
{{1,8},{2,4,5,6,7},{3}}
=> ?
=> ? = 5
{{1,5},{2,3,4},{6,7,8}}
=> ?
=> ? = 3
{{1,3},{2,4,6,7},{5},{8}}
=> ?
=> ? = 4
{{1,6},{2,3,4},{5,8},{7}}
=> ?
=> ? = 3
{{1,6},{2,3,4},{5,7},{8}}
=> ?
=> ? = 3
{{1,3},{2,5,7},{4,6},{8}}
=> ?
=> ? = 3
{{1,3},{2,4,7},{5,6},{8}}
=> ?
=> ? = 3
{{1,4},{2,7},{3,5,6},{8}}
=> ?
=> ? = 3
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> 2
{{1},{2}}
=> [1,1]
=> [2]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 2
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 2
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 2
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 2
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 3
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> 2
{{1},{2},{3,4},{5,6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1},{2},{3,4},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1},{2},{3,4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1},{2},{3,7,8},{4},{5},{6}}
=> ?
=> ?
=> ? = 3
{{1},{2},{3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 4
{{1},{2,3},{4},{5},{6,7},{8}}
=> ?
=> ?
=> ? = 2
{{1},{2,3},{4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1},{2,4},{3},{5,7},{6},{8}}
=> ?
=> ?
=> ? = 2
{{1},{2,7,8},{3},{4},{5},{6}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,4},{5},{6,8},{7}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,4},{5,6,7},{8}}
=> ?
=> ?
=> ? = 3
{{1},{2,7,8},{3,5,6},{4}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? = 6
{{1},{2,8},{3,4,7},{5},{6}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? = 6
{{1},{2,8},{3,4,5,6,7}}
=> ?
=> ?
=> ? = 5
{{1,2},{3},{4},{5},{6,7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4},{5},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,2},{3},{4},{5,6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4,5},{6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4,8},{5},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,4},{5},{6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,5},{4},{6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,6},{4},{5},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,8},{4},{5},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,4,5,7},{6},{8}}
=> ?
=> ?
=> ? = 4
{{1,4},{2},{3},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1,5},{2},{3},{4},{6},{7,8}}
=> ?
=> ?
=> ? = 2
{{1,8},{2},{3},{4},{5,6},{7}}
=> ?
=> ?
=> ? = 2
{{1,5},{2},{3,4},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,8},{2},{3,4,5,6,7}}
=> ?
=> ?
=> ? = 5
{{1,4},{2,3},{5},{6},{7,8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,7},{8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,8},{7}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,4},{2,3},{5,6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5,7},{6},{8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5,6,7},{8}}
=> ?
=> ?
=> ? = 3
{{1,6},{2,3},{4},{5},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,8},{2,4,5,6,7},{3}}
=> ?
=> ?
=> ? = 5
{{1,5},{2,3,4},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,3},{2,4,6,7},{5},{8}}
=> ?
=> ?
=> ? = 4
{{1,6},{2,3,4},{5,8},{7}}
=> ?
=> ?
=> ? = 3
{{1,6},{2,3,4},{5,7},{8}}
=> ?
=> ?
=> ? = 3
{{1,3},{2,5,7},{4,6},{8}}
=> ?
=> ?
=> ? = 3
{{1,3},{2,4,7},{5,6},{8}}
=> ?
=> ?
=> ? = 3
{{1,4},{2,7},{3,5,6},{8}}
=> ?
=> ?
=> ? = 3
Description
The length of the partition.
Matching statistic: St000734
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> 2
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> 3
{{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
{{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
{{1},{2},{3,4},{5,6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1},{2},{3,4},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1},{2},{3,4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1},{2},{3,7,8},{4},{5},{6}}
=> ?
=> ?
=> ? = 3
{{1},{2},{3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 4
{{1},{2,3},{4},{5},{6,7},{8}}
=> ?
=> ?
=> ? = 2
{{1},{2,3},{4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1},{2,4},{3},{5,7},{6},{8}}
=> ?
=> ?
=> ? = 2
{{1},{2,7,8},{3},{4},{5},{6}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,4},{5},{6,8},{7}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,4},{5,6,7},{8}}
=> ?
=> ?
=> ? = 3
{{1},{2,7,8},{3,5,6},{4}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? = 6
{{1},{2,8},{3,4,7},{5},{6}}
=> ?
=> ?
=> ? = 3
{{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? = 6
{{1},{2,8},{3,4,5,6,7}}
=> ?
=> ?
=> ? = 5
{{1,2},{3},{4},{5},{6,7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4},{5},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,2},{3},{4},{5,6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4,5},{6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3},{4,8},{5},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,4},{5},{6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,5},{4},{6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,6},{4},{5},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,8},{4},{5},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,2},{3,4,5,7},{6},{8}}
=> ?
=> ?
=> ? = 4
{{1,4},{2},{3},{5,8},{6,7}}
=> ?
=> ?
=> ? = 2
{{1,5},{2},{3},{4},{6},{7,8}}
=> ?
=> ?
=> ? = 2
{{1,8},{2},{3},{4},{5,6},{7}}
=> ?
=> ?
=> ? = 2
{{1,5},{2},{3,4},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,8},{2},{3,4,5,6,7}}
=> ?
=> ?
=> ? = 5
{{1,4},{2,3},{5},{6},{7,8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,7},{8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,8},{7}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,4},{2,3},{5,6},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5,7},{6},{8}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5,8},{6},{7}}
=> ?
=> ?
=> ? = 2
{{1,4},{2,3},{5,6,7},{8}}
=> ?
=> ?
=> ? = 3
{{1,6},{2,3},{4},{5},{7},{8}}
=> ?
=> ?
=> ? = 2
{{1,8},{2,4,5,6,7},{3}}
=> ?
=> ?
=> ? = 5
{{1,5},{2,3,4},{6,7,8}}
=> ?
=> ?
=> ? = 3
{{1,3},{2,4,6,7},{5},{8}}
=> ?
=> ?
=> ? = 4
{{1,6},{2,3,4},{5,8},{7}}
=> ?
=> ?
=> ? = 3
{{1,6},{2,3,4},{5,7},{8}}
=> ?
=> ?
=> ? = 3
{{1,3},{2,5,7},{4,6},{8}}
=> ?
=> ?
=> ? = 3
{{1,3},{2,4,7},{5,6},{8}}
=> ?
=> ?
=> ? = 3
{{1,4},{2,7},{3,5,6},{8}}
=> ?
=> ?
=> ? = 3
Description
The last entry in the first row of a standard tableau.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000141The maximum drop size of a permutation. St000288The number of ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000691The number of changes of a binary word. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000808The number of up steps of the associated bargraph. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000676The number of odd rises of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000439The position of the first down step of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000874The position of the last double rise in a Dyck path. St000025The number of initial rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000653The last descent of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St000383The last part of an integer composition. St000031The number of cycles in the cycle decomposition of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000306The bounce count of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000662The staircase size of the code of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000007The number of saliances of the permutation. St000308The height of the tree associated to a permutation. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000990The first ascent of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000062The length of the longest increasing subsequence of the permutation. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000166The depth minus 1 of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000094The depth of an ordered tree. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000028The number of stack-sorts needed to sort a permutation. St001330The hat guessing number of a graph. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!