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Your data matches 313 different statistics following compositions of up to 3 maps.
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Matching statistic: St000397
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Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1
[1,0,1,0]
=> [[],[]]
=> 2
[1,1,0,0]
=> [[[]]]
=> 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 2
[1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 2
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 2
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2
Description
The Strahler number of a rooted tree.
Matching statistic: St000326
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> ?
=> ? => ? = 1 - 1
[1,0,1,0]
=> [1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,0,0]
=> []
=> ?
=> ? => ? = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 1110 => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 110 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1001110 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 10100 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 10110 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1010 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 100110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 11110 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 10010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1110 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1000 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 110 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> 101101110 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 1000001110 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> 110111110 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> 100111110 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> 1011100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 10000100 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> 1110110 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1010110 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 111010 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 101010 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 101110 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 10000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> 10110110 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 100000110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> 11011110 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,1,1,1,1]
=> 10011110 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,2,2,1]
=> 1011010 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 10000010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,2,1,1,1]
=> 1101110 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1001110 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,2,2]
=> 11100 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,2]
=> 10100 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,1,1]
=> 10110 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1010 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3,2,2]
=> 101100 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> 1000000 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 110110 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [3,1,1]
=> 100110 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [5,3,2,2,1,1,1]
=> 100101101110 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [3,2,2,2,2,1,1,1]
=> 10111101110 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [6,5,1,1,1]
=> 10100001110 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [6,2,2,1,1,1]
=> 100001101110 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [6,3,1,1,1]
=> 10001001110 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [5,2,2,1,1,1,1,1]
=> 1000110111110 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [2,2,2,2,1,1,1,1,1]
=> 11110111110 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1,1,1,1,1]
=> 10110111110 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> 1000011111110 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,3,2,2,2]
=> 1001011100 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [6,2,2,2]
=> 1000011100 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [5,2,2,2,1,1]
=> 10001110110 => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [5,3,2,2,1,1]
=> 10010110110 => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,2,2,2,2,1,1]
=> 1011110110 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [6,5,1,1]
=> 1010000110 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [6,2,2,1,1]
=> 10000110110 => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,3,1,1]
=> 1000100110 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [5,2,2,1,1,1,1]
=> 100011011110 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [5,3,1,1,1,1]
=> 10010011110 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [5,1,1,1,1,1,1]
=> 100001111110 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [6,2,2,1]
=> 1000011010 => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [5,2,2,1,1,1]
=> 10001101110 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [6,2,2,2,2,1,1,1]
=> 10000111101110 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> [6,5,3,1,1,1]
=> 101001001110 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> [6,3,2,2,1,1,1]
=> 1000101101110 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [12,1,1,1]
=> 1000000000001110 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 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: St000296
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0]
=> [1]
=> [1]
=> 10 => 0 = 2 - 2
[1,1,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 1110 => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 100 => 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 110 => 0 = 2 - 2
[1,1,0,1,0,0]
=> [1]
=> [1]
=> 10 => 0 = 2 - 2
[1,1,1,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1001110 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 10100 => 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1010 => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 100110 => 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 11110 => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 10010 => 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1110 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 100 => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1000 => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 110 => 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 10 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> 101101110 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 1000001110 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> 110111110 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> 100111110 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> 1011100 => 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 10000100 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> 1110110 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1010110 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 111010 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 101010 => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 10000 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> 10110110 => 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 100000110 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> 11011110 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,1,1,1,1]
=> 10011110 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> 1111110 => 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,2,2,1]
=> 1011010 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 10000010 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,2,1,1,1]
=> 1101110 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1001110 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,2,2]
=> 11100 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,2]
=> 10100 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1010 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3,2,2]
=> 101100 => 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> 1000000 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 110110 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [3,1,1]
=> 100110 => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [5,3,2,2,1,1,1]
=> 100101101110 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [3,2,2,2,2,1,1,1]
=> 10111101110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [6,5,1,1,1]
=> 10100001110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [6,2,2,1,1,1]
=> 100001101110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [6,3,1,1,1]
=> 10001001110 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [5,2,2,1,1,1,1,1]
=> 1000110111110 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [2,2,2,2,1,1,1,1,1]
=> 11110111110 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1,1,1,1,1]
=> 10110111110 => ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> 1000011111110 => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,3,2,2,2]
=> 1001011100 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [6,2,2,2]
=> 1000011100 => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [5,2,2,2,1,1]
=> 10001110110 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [5,3,2,2,1,1]
=> 10010110110 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,2,2,2,2,1,1]
=> 1011110110 => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [6,5,1,1]
=> 1010000110 => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [6,2,2,1,1]
=> 10000110110 => ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,3,1,1]
=> 1000100110 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [5,2,2,1,1,1,1]
=> 100011011110 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [5,3,1,1,1,1]
=> 10010011110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [5,1,1,1,1,1,1]
=> 100001111110 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [6,2,2,1]
=> 1000011010 => ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [5,2,2,1,1,1]
=> 10001101110 => ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [6,2,2,2,2,1,1,1]
=> 10000111101110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> [6,5,3,1,1,1]
=> 101001001110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> [6,3,2,2,1,1,1]
=> 1000101101110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [12,1,1,1]
=> 1000000000001110 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000396
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,1,1,0,0,0]
=> [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]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> 2
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,[.,.]],.]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],[[.,.],.]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [[[.,.],.],[[[.,[.,.]],.],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[.,[[.,.],.]]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[[.,.],.],[[.,[.,.]],[.,[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[.,[[.,.],[.,.]]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],.],[[[.,.],[.,.]],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[[.,.],.],[[.,[.,[.,.]]],[.,.]]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],.],[.,[[.,[.,.]],[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[.,[[[[.,.],.],.],.]],.],[.,.]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[[.,[[[.,.],.],.]],.],[.,[.,.]]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[.,[[[.,[.,.]],.],.]],.],[.,.]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[[.,[[.,.],.]],.],[[.,.],[.,.]]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],[.,[.,[.,.]]]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[.,[[.,[[.,.],.]],.]],.],[.,.]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[[.,[.,.]],.]],.],[.,[.,.]]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[.,[[[.,.],[.,.]],.]],.],[.,.]]
=> ? = 2
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St001568
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> ? = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2 - 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001696
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,0,1,0]
=> [1]
=> [1]
=> [[1]]
=> 0 = 2 - 2
[1,1,0,0]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [[1,2]]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [[1]]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> [[1,2],[3]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [[1,2]]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> [[1,2,3]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [[1]]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> [[1,2],[3,4]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> [[1,2,3,4,5,6],[7]]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> [[1,2],[3]]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [5,3,1,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13],[14],[15]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [8,3,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12],[13],[14]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [6,5,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12],[13],[14]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [6,1,1,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [6,3,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11],[12]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [5,4,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12],[13],[14]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [8,4,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12],[13]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [4,3,1,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11],[12]]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,3,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12],[13],[14]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [6,2,1,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [5,2,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12],[13]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,2,1,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11],[12]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [5,2,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13],[14]]
=> ? = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [8,3,1,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [6,5,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12],[13]]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [6,1,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [5,4,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12],[13]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [8,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [5,1,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [5,3,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12]]
=> ? = 2 - 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ?
=> ?
=> ? = 2 - 2
Description
The natural major index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000687
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0]
=> [1,0,1,0]
=> [1]
=> [1,0]
=> 0 = 2 - 2
[1,1,0,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
[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]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 2
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules.
This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St001371
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0]
=> [1]
=> [1]
=> 10 => 0 = 2 - 2
[1,1,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 1110 => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 100 => 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 110 => 0 = 2 - 2
[1,1,0,1,0,0]
=> [1]
=> [1]
=> 10 => 0 = 2 - 2
[1,1,1,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1001110 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 10100 => 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1010 => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 100110 => 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 11110 => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 10010 => 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1110 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 100 => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1000 => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 110 => 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 10 => 0 = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> 101101110 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 1000001110 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> 110111110 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> 100111110 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> 1011100 => 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 10000100 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> 1110110 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1010110 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 111010 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 101010 => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 10000 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> 10110110 => 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 100000110 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> 11011110 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,1,1,1,1]
=> 10011110 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> 1111110 => 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,2,2,1]
=> 1011010 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 10000010 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,2,1,1,1]
=> 1101110 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1001110 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,2,2]
=> 11100 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,2]
=> 10100 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1010 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3,2,2]
=> 101100 => 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> 1000000 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 110110 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [3,1,1]
=> 100110 => 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,1]
=> 11110 => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [5,3,2,2,1,1,1]
=> 100101101110 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [3,2,2,2,2,1,1,1]
=> 10111101110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [6,5,1,1,1]
=> 10100001110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [6,2,2,1,1,1]
=> 100001101110 => ? = 2 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [6,3,1,1,1]
=> 10001001110 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [5,2,2,1,1,1,1,1]
=> 1000110111110 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [2,2,2,2,1,1,1,1,1]
=> 11110111110 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1,1,1,1,1]
=> 10110111110 => ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> 1000011111110 => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => ? = 2 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => ? = 2 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,3,2,2,2]
=> 1001011100 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [6,2,2,2]
=> 1000011100 => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [5,2,2,2,1,1]
=> 10001110110 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [5,3,2,1,1]
=> 1001010110 => ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [6,2,1,1]
=> 1000010110 => ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [5,2,1,1,1,1]
=> 10001011110 => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [5,2,2,2,1]
=> 1000111010 => ? = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [5,2,1,1,1]
=> 1000101110 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [5,3,2,2,1,1]
=> 10010110110 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,2,2,2,2,1,1]
=> 1011110110 => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [6,5,1,1]
=> 1010000110 => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [6,2,2,1,1]
=> 10000110110 => ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,3,1,1]
=> 1000100110 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [5,2,2,1,1,1,1]
=> 100011011110 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [5,3,1,1,1,1]
=> 10010011110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [6,1,1,1,1]
=> 10000011110 => ? = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [5,1,1,1,1,1,1]
=> 100001111110 => ? = 2 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [2,2,1,1,1,1,1,1]
=> 1101111110 => ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [3,1,1,1,1,1,1]
=> 1001111110 => ? = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [5,3,2,2,1]
=> 1001011010 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [6,2,2,1]
=> 1000011010 => ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [5,2,2,1,1,1]
=> 10001101110 => ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [5,3,1,1,1]
=> 1001001110 => ? = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 1000001110 => ? = 2 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [5,1,1,1,1,1]
=> 10000111110 => ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [5,2,2,1,1]
=> 1000110110 => ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [5,1,1,1,1]
=> 1000011110 => ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? => ? = 1 - 2
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St000629
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> []
=> => ? = 1 - 2
[1,0,1,0]
=> [1]
=> [[1]]
=> => ? = 2 - 2
[1,1,0,0]
=> []
=> []
=> => ? = 1 - 2
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 0 = 2 - 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 1 => 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 0 => 0 = 2 - 2
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> => ? = 2 - 2
[1,1,1,0,0,0]
=> []
=> []
=> => ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0 = 2 - 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0 = 2 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 0 = 2 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 0 = 2 - 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1 => 0 = 2 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 00 => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 0 => 0 = 2 - 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> => ? = 2 - 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> 101001000 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> 10100100 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> 10101000 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 1010100 => 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 101010 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> 11001000 => 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 1100100 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 1101000 => 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 110100 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 11010 => 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 111000 => 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> 01001000 => 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0100100 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0101000 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 010100 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 01010 => 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 1001000 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 100100 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 101000 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 001000 => 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 00100 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 01000 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 0 = 2 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> => ? = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> => ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 1010010001000 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> 1010010010000 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> 101001001000 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> 1010100010000 => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> 101010001000 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> 10101001000 => ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> 1010100100 => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> 10101010000 => ? = 2 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> 1010101000 => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> 1100100010000 => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> 110010001000 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> 110010010000 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> 11001001000 => ? = 2 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> 1100100100 => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> 110100010000 => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> 11010001000 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> 11010010000 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> 1101001000 => ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> 1101010000 => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> 11100010000 => ? = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> 1110001000 => ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> 1110010000 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> 0100100010000 => ? = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> 010010001000 => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> 010010010000 => ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> 0100100100 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> 010100010000 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> 01010010000 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> 0101001000 => ? = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> 0101010000 => ? = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
=> 100100010000 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> 10010010000 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> 1001001000 => ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> 10100010000 => ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> 1010001000 => ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> 1010010000 => ? = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> 1100010000 => ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> 0100010000 => ? = 2 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1]]
=> => ? = 2 - 2
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[1,1,0,0]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,2,3]]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,2,3]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,2,3,4,5],[6,7],[8]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,2,3,4],[5,6],[7]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,2,3,4,5,6],[7],[8]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,2,3,4,5,6],[7],[8]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,2,3,4,5,6],[7,8]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,2,3,4,5,6],[7,8]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,2,3,4,5,6,7],[8]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12],[13]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [[1,2,3,4,5,6,7,8,9],[10,11],[12,13],[14]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [[1,2,3,4,5,6,7],[8,9],[10,11],[12]]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,2,3,4,5,6,7],[8,9],[10,11],[12]]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12],[13],[14]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12],[13]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [[1,2,3,4,5,6,7],[8,9,10],[11],[12]]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [[1,2,3,4,5,6,7,8,9],[10,11],[12],[13]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11],[12]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11],[12]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [[1,2,3,4,5,6,7,8,9],[10,11],[12,13]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12],[13]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [[1,2,3,4,5,6,7,8,9],[10,11],[12]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15,16,17],[18,19],[20]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13,14],[15,16,17],[18,19],[20]]
=> ?
=> ? = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
The following 303 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000842The breadth of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St000769The major index of a composition regarded as a word. St000768The number of peaks in an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000766The number of inversions of an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000862The number of parts of the shifted shape of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000834The number of right outer peaks of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000805The number of peaks of the associated bargraph. St000761The number of ascents in an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000679The pruning number of an ordered tree. St000535The rank-width of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000058The order of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000317The cycle descent number of a permutation. St000674The number of hills of a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000570The Edelman-Greene number of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. 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 $eA$. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001204Call 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 special CNakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001545The second Elser number of a connected graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000091The descent variation of a composition. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000028The number of stack-sorts needed to sort a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001737The number of descents of type 2 in a permutation. St000920The logarithmic height of a Dyck path. St001330The hat guessing number of a graph. St000647The number of big descents of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000255The number of reduced Kogan faces with the permutation as type. St000374The number of exclusive right-to-left minima of a permutation. St001735The number of permutations with the same set of runs. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001665The number of pure excedances of a permutation. St000845The maximal number of elements covered by an element in a poset. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000914The sum of the values of the Möbius function of a poset. St000245The number of ascents of a permutation. St000183The side length of the Durfee square of an integer partition. St001335The cardinality of a minimal cycle-isolating set of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000035The number of left outer peaks of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000352The Elizalde-Pak rank of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000638The number of up-down runs of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000700The protection number of an ordered tree. St000783The side length of the largest staircase partition fitting into a partition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001734The lettericity of a graph. St001962The proper pathwidth of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000353The number of inner valleys of a permutation. St000472The sum of the ascent bottoms of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000779The tier of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001112The 3-weak dynamic number of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001469The holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001638The book thickness of a graph. St001110The 3-dynamic chromatic number of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001555The order of a signed permutation. St001730The number of times the path corresponding to a binary word crosses the base line. 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 $eA$. St000306The bounce count of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000381The largest part of an integer composition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001870The number of positive entries followed by a negative entry in a signed permutation. St000141The maximum drop size of a permutation. St000662The staircase size of the code of a permutation. St001060The distinguishing index of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000891The number of distinct diagonal sums of a permutation matrix. St001624The breadth of a lattice. St000015The number of peaks of a Dyck path. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000903The number of different parts of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001202Call 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 special CNakayama algebra. 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:
St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001569The maximal modular displacement of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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