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Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
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Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00109: Permutations ādescent wordā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 0
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 0
[1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0100 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000392
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
Mp00094: Integer compositions āto binary wordā¶ Binary words
St000392: Binary words ā¶ ā¤Result quality: 85% āvalues known / values provided: 85%ādistinct values known / distinct values provided: 100%
Mp00071: Permutations ādescent compositionā¶ Integer compositions
Mp00094: Integer compositions āto binary wordā¶ Binary words
St000392: Binary words ā¶ ā¤Result quality: 85% āvalues known / values provided: 85%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2] => 10 => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3] => 100 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1] => 101 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,2] => 110 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 101 => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,2] => 110 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => 1001 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => 1010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => 1001 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,2] => 1010 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => 1100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => 1101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => 1010 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1] => 1001 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,2] => 1010 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3] => 1100 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [1,2,1] => 1101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2] => 1010 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,3] => 1100 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1] => 10001 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [3,2] => 10010 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1] => 10001 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,2] => 10010 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3] => 10100 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1] => 10101 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,2] => 10010 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => 10001 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2] => 10010 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,3] => 10100 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1] => 10101 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2] => 10010 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,3] => 10100 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,4] => 11000 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,1] => 11001 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,2] => 11010 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => 11001 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,2,2] => 11010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3] => 10100 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => 10101 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => 10010 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1] => 10001 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2] => 10010 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,3] => 10100 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => 10101 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2] => 10010 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,3] => 10100 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [1,4] => 11000 => 2 = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [1,9] => 1100000000 => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,2,3,4,5,6,7,8,9] => [2,8] => 1010000000 => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [1,10] => 11000000000 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [2,8] => 1010000000 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [3,7] => 1001000000 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [2,9] => 10100000000 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,10,3,4,5,6,7,8] => [2,2,6] => 1010100000 => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,10,2,3,4,5,6,7] => [4,6] => 1000100000 => ? = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,9,1,2,3,4,5,10] => [4,6] => 1000100000 => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,7,8,9,10,2,3,4,5,6] => [5,5] => 1000010000 => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,1,2,3,4,5,6,7,10,9] => [1,8,1] => 1100000001 => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,10,3,4,5,6,7,8,9] => [1,2,7] => 1101000000 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7,9,10] => [1,9] => 1100000000 => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,9,2,3,4,5,6,7,8,10] => [2,8] => 1010000000 => ? = 0 + 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,10,3,4,5,6,7,8,9] => [3,7] => 1001000000 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,2] => [9,1] => 1000000001 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [9,1] => 1000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [10,1] => 10000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [10,1] => 10000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,6,9,7,10,8] => [7,2,1] => 1000000101 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [1,8,1] => 1100000001 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [10,1] => 10000000001 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [10,1] => 10000000001 => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,8,9,10,2] => [2,7,1] => 1010000001 => ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,9,10,3] => [1,8,1] => 1100000001 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,12,2] => [11,1] => 100000000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,9,12,10,11] => [10,2] => 100000000010 => ? = 0 + 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,6,7,8,9,10,2] => [3,6,1] => 1001000001 => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,8,9,10,11,2] => [2,8,1] => 10100000001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5,10,6,7,8,9] => [6,4] => 1000001000 => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,7,8,9,10,2] => [4,5,1] => 1000100001 => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,1,2,3,6,7,8,9,10,4] => [1,8,1] => 1100000001 => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,8,9,10,2] => [5,4,1] => 1000010001 => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,9,10,2] => [6,3,1] => 1000001001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,10,5,6,7,8,9] => [5,5] => 1000010000 => ? = 0 + 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,7,8,9,10,4] => [2,7,1] => 1010000001 => ? = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,1,2,3,4,7,8,9,10,5] => [1,8,1] => 1100000001 => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,7,8,9,1,10,2] => [7,2,1] => 1000000101 => ? = 0 + 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,7,8,9,1,2,10,3] => [6,3,1] => 1000001001 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7,10,9] => [7,2,1] => 1000000101 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,6,8,9,10] => [6,4] => 1000001000 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8,9,10] => [5,5] => 1000010000 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7,10,9] => [5,2,2,1] => 1000010101 => ? = 0 + 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000382
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00069: Permutations ācomplementā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 85% āvalues known / values provided: 85%ādistinct values known / distinct values provided: 100%
Mp00069: Permutations ācomplementā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 85% āvalues known / values provided: 85%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1,2] => [2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [1,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => [1,1,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => [1,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [4,1,3,2] => [1,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => [1,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [1,1,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => [2,1,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [5,1,4,3,2] => [1,2,1,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [1,2,1,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [1,2,2] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,2] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => [1,2,1,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,2,2] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => [1,2,1,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,4,6,5,8,7] => [8,6,7,5,3,4,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,3,2,4,7,5,8,6] => [8,6,7,5,2,4,1,3] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,6,7,4,5,8] => [8,6,7,3,2,5,4,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,6,7,4,8,5] => [8,6,7,3,2,5,1,4] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,6,8,4,5,7] => [8,6,7,3,1,5,4,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => [8,6,5,4,7,3,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,4,6,2,5,8,7] => [8,6,5,3,7,4,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,6,7,2,5,8] => [8,6,5,3,2,7,4,1] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,5,6,8,2,4,7] => [8,6,4,3,1,7,5,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,3,6,8,7] => [8,5,7,4,6,3,1,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,4,5,2,3,7,6,8] => [8,5,4,7,6,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,4,5,2,3,8,6,7] => [8,5,4,7,6,1,3,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,7,3,8] => [8,5,4,7,3,2,6,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,4,6,2,3,5,7,8] => [8,5,3,7,6,4,2,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,2,6,7,3,4,8] => [8,4,7,3,2,6,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,2,6,7,8,3,4] => [8,4,7,3,2,1,6,5] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => [8,4,3,7,2,6,5,1] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,4,7,5,8,6] => [7,8,6,5,2,4,1,3] => ? => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,4,5,8,7] => [7,8,6,3,5,4,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6,8,7] => [7,8,5,4,6,3,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,6,3,4,8,5,7] => [7,8,3,6,5,1,4,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,1,6,7,3,4,8,5] => [7,8,3,2,6,5,1,4] => ? => ? = 1 + 1
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,1,7,8,4,5,6] => [7,6,8,2,1,5,4,3] => ? => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,3,5,7,8,1,4,6] => [7,6,4,2,1,8,5,3] => ? => ? = 0 + 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,1,4,5,7] => [7,6,3,1,8,5,4,2] => ? => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [2,4,5,1,3,7,6,8] => [7,5,4,8,6,2,3,1] => ? => ? = 0 + 1
[1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,5,1,7,8,3,4,6] => [7,4,8,2,1,6,5,3] => ? => ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,4,2,5,8,6,7] => [6,8,5,7,4,1,3,2] => ? => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,5,2,7,8,6] => [6,8,5,4,7,2,1,3] => ? => ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5,7,8] => [6,8,3,7,5,4,2,1] => ? => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> [4,5,1,2,6,8,3,7] => [5,4,8,7,3,1,6,2] => ? => ? = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [4,5,1,6,2,3,7,8] => [5,4,8,3,7,6,2,1] => ? => ? = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,6,1,2,7,3,8] => [5,4,3,8,7,2,6,1] => ? => ? = 0 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [4,6,7,1,2,3,8,5] => [5,3,2,8,7,6,1,4] => ? => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [2,11,10,9,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [11,1,10,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1,1] => ? = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,1,2,3,4,5,9] => [4,3,2,9,8,7,6,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,2,10,3,4,5,6,7,8] => [10,2,9,1,8,7,6,5,4,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,8,2,3,4,5,9,6,7] => [9,2,8,7,6,5,1,4,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,7,2,8,9,3,4,5,6] => [9,3,8,2,1,7,6,5,4] => ? => ? = 0 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,9,1,2,3,4,5,10] => [5,4,3,2,10,9,8,7,6,1] => ? => ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,7,1,8,2,3,4,9] => [5,4,3,9,2,8,7,6,1] => ? => ? = 0 + 1
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,8,1,2,3,5,9] => [6,4,3,2,9,8,7,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,8,2,9,3,4,5] => [9,4,3,2,8,1,7,6,5] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,5,7,8,9,2,3,4,6] => [9,5,3,2,1,8,7,6,4] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,6,2,7,8,9,3,4,5] => [9,4,8,3,2,1,7,6,5] => ? => ? = 0 + 1
Description
The first part of an integer composition.
Matching statistic: St000745
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 83% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 100%
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 83% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [[1,2]]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,1,6,7,3,8,4,5] => ?
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,3,6,7,1,4,5,8] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,6,7,1,4,8,5] => ?
=> ? = 0 + 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,1,4,5,7] => ?
=> ? = 0 + 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,1,3,8,6] => ?
=> ? = 0 + 1
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,6,1,3,7,8,5] => ?
=> ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => ?
=> ? = 0 + 1
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [2,4,6,7,1,8,3,5] => ?
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,8,4,7] => ?
=> ? = 0 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [3,1,2,5,6,4,7,8] => ?
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,1,2,5,7,8,4,6] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,4,7,2,8,5,6] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,5,2,4,6,7,8] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,5,2,4,6,8,7] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,5,2,4,7,8,6] => ?
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,4,1,7,2,5,8,6] => ?
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0,1,0]
=> [4,1,5,7,2,3,6,8] => ?
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [4,1,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> [4,5,1,7,2,3,6,8] => ?
=> ? = 0 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [4,5,7,1,2,3,8,6] => ?
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [5,6,1,2,7,3,4,8] => ?
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [[1,2,3,6,7,8,9,10],[4,5]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [[1,2,4,5,6,7,8,9,10,11],[3]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,10,2,3,4,5,6,7] => [[1,2,3,4,8,9,10],[5,6,7]]
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,7,1,8,2,3,4,9] => [[1,2,3,5,9],[4,6,7,8]]
=> ? = 0 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5,9,8] => [[1,2,5,6,7,8],[3,4,9]]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,8,2,9,3,4,5] => [[1,2,3,4,6],[5,7,8,9]]
=> ? = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000877
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00069: Permutations ācomplementā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000877: Binary words ā¶ ā¤Result quality: 81% āvalues known / values provided: 81%ādistinct values known / distinct values provided: 100%
Mp00069: Permutations ācomplementā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000877: Binary words ā¶ ā¤Result quality: 81% āvalues known / values provided: 81%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => 1 => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 11 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 10 => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => 01 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 10 => 0
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 111 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => 110 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 101 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => 110 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [4,1,3,2] => 101 => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 011 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => 101 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => 110 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 101 => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => 011 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => 101 => 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => 1110 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => 1101 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => 1110 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => 1011 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => 1101 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => 1101 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,2,4,3,1] => 1011 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,2,4,1,3] => 1010 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [5,2,1,4,3] => 1101 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [5,1,4,3,2] => 1011 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => 0101 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => 0110 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => 0101 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 1011 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 1010 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 1101 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 1101 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => 1101 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,5,6,4,8,7] => [8,7,6,4,3,5,1,2] => ? => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,7,5,6,8] => [8,7,5,6,2,4,3,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [8,7,5,4,6,2,1,3] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,6,3,8,7] => [8,7,5,4,3,6,1,2] => ? => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,2,4,5,7,3,6,8] => [8,7,5,4,2,6,3,1] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,3,5,8,6] => [8,7,5,2,6,4,1,3] => ? => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,3,8,5,6] => [8,7,5,2,6,1,4,3] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,3,4,5,8,7] => [8,7,3,6,5,4,1,2] => ? => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,3,4,5,8] => [8,7,3,2,6,5,4,1] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,4,6,5,8,7] => [8,6,7,5,3,4,1,2] => ? => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,3,2,4,7,5,8,6] => [8,6,7,5,2,4,1,3] => ? => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [8,6,7,4,3,5,1,2] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,6,7,4,5,8] => [8,6,7,3,2,5,4,1] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,6,7,4,8,5] => [8,6,7,3,2,5,1,4] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,2,6,7,8,4,5] => [8,6,7,3,2,1,5,4] => ? => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,6,8,4,5,7] => [8,6,7,3,1,5,4,2] => ? => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => [8,6,5,4,7,3,1,2] => ? => ? = 0
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,8,2,6,7] => [8,6,5,4,1,7,3,2] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,4,6,2,5,7,8] => [8,6,5,3,7,4,2,1] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,4,6,2,5,8,7] => [8,6,5,3,7,4,1,2] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,6,7,2,5,8] => [8,6,5,3,2,7,4,1] => ? => ? = 0
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,8,2,5,7] => [8,6,5,3,1,7,4,2] => ? => ? = 0
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,5,6,8,2,4,7] => [8,6,4,3,1,7,5,2] => ? => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,2,4,5,7,8] => [8,6,3,7,5,4,2,1] => ? => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,6,2,7,4,5,8] => [8,6,3,7,2,5,4,1] => ? => ? = 0
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,7,8,2,4,5,6] => [8,6,2,1,7,5,4,3] => ? => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,2,3,6,5,7,8] => [8,5,7,6,3,4,2,1] => ? => ? = 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,4,2,3,7,5,8,6] => [8,5,7,6,2,4,1,3] => ? => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,3,6,8,7] => [8,5,7,4,6,3,1,2] => ? => ? = 0
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,4,2,6,7,3,5,8] => [8,5,7,3,2,6,4,1] => ? => ? = 0
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,7,3,5,8,6] => [8,5,7,2,6,4,1,3] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,4,5,2,3,7,6,8] => [8,5,4,7,6,2,3,1] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,4,5,2,3,8,6,7] => [8,5,4,7,6,1,3,2] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,7,3,8] => [8,5,4,7,3,2,6,1] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,6,8,3,7] => [8,5,4,7,3,1,6,2] => ? => ? = 0
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,7,8,3,6] => [8,5,4,7,2,1,6,3] => ? => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => [8,5,4,3,7,1,6,2] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,4,6,2,3,5,7,8] => [8,5,3,7,6,4,2,1] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,2,3,7,8,5] => [8,5,3,7,6,2,1,4] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,4,6,7,8,2,3,5] => [8,5,3,2,1,7,6,4] => ? => ? = 0
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [8,5,3,1,7,6,4,2] => ? => ? = 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,7,2,3,5,8,6] => [8,5,2,7,6,4,1,3] => ? => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ? = 0
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,5,2,3,8,4,6,7] => [8,4,7,6,1,5,3,2] => ? => ? = 0
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,2,6,7,3,4,8] => [8,4,7,3,2,6,5,1] => ? => ? = 0
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,2,6,7,8,3,4] => [8,4,7,3,2,1,6,5] => ? => ? = 0
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => [8,4,3,7,2,6,5,1] => ? => ? = 0
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000326
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00069: Permutations ācomplementā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 81% āvalues known / values provided: 81%ādistinct values known / distinct values provided: 100%
Mp00069: Permutations ācomplementā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 81% āvalues known / values provided: 81%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => 1 => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 11 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 10 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => 01 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 10 => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => 01 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 111 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => 110 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 101 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => 110 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [4,1,3,2] => 101 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 011 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => 101 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => 110 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 101 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => 011 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 010 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => 101 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => 1110 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => 1101 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => 1110 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => 1011 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => 1101 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => 1101 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,2,4,3,1] => 1011 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,2,4,1,3] => 1010 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [5,2,1,4,3] => 1101 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [5,1,4,3,2] => 1011 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => 0101 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => 0110 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => 0101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 1011 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 1010 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 1101 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 1101 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => 1101 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,5,6,4,8,7] => [8,7,6,4,3,5,1,2] => ? => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,7,5,6,8] => [8,7,5,6,2,4,3,1] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,2,4,5,3,7,8,6] => [8,7,5,4,6,2,1,3] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,6,3,8,7] => [8,7,5,4,3,6,1,2] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,2,4,5,7,3,6,8] => [8,7,5,4,2,6,3,1] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,3,5,8,6] => [8,7,5,2,6,4,1,3] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,3,8,5,6] => [8,7,5,2,6,1,4,3] => ? => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,3,4,5,8,7] => [8,7,3,6,5,4,1,2] => ? => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,3,4,5,8] => [8,7,3,2,6,5,4,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,4,6,5,8,7] => [8,6,7,5,3,4,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,3,2,4,7,5,8,6] => [8,6,7,5,2,4,1,3] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [8,6,7,4,3,5,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,6,7,4,5,8] => [8,6,7,3,2,5,4,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,2,6,7,4,8,5] => [8,6,7,3,2,5,1,4] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,2,6,7,8,4,5] => [8,6,7,3,2,1,5,4] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,6,8,4,5,7] => [8,6,7,3,1,5,4,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => [8,6,5,4,7,3,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,8,2,6,7] => [8,6,5,4,1,7,3,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,4,6,2,5,7,8] => [8,6,5,3,7,4,2,1] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,4,6,2,5,8,7] => [8,6,5,3,7,4,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,6,7,2,5,8] => [8,6,5,3,2,7,4,1] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,8,2,5,7] => [8,6,5,3,1,7,4,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,5,6,8,2,4,7] => [8,6,4,3,1,7,5,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,2,4,5,7,8] => [8,6,3,7,5,4,2,1] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,3,6,2,7,4,5,8] => [8,6,3,7,2,5,4,1] => ? => ? = 0 + 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,7,8,2,4,5,6] => [8,6,2,1,7,5,4,3] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,2,3,6,5,7,8] => [8,5,7,6,3,4,2,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,4,2,3,7,5,8,6] => [8,5,7,6,2,4,1,3] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,3,6,8,7] => [8,5,7,4,6,3,1,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,4,2,6,7,3,5,8] => [8,5,7,3,2,6,4,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,7,3,5,8,6] => [8,5,7,2,6,4,1,3] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,4,5,2,3,7,6,8] => [8,5,4,7,6,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,4,5,2,3,8,6,7] => [8,5,4,7,6,1,3,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,7,3,8] => [8,5,4,7,3,2,6,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,6,8,3,7] => [8,5,4,7,3,1,6,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,7,8,3,6] => [8,5,4,7,2,1,6,3] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => [8,5,4,3,7,1,6,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,4,6,2,3,5,7,8] => [8,5,3,7,6,4,2,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,2,3,7,8,5] => [8,5,3,7,6,2,1,4] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,4,6,7,8,2,3,5] => [8,5,3,2,1,7,6,4] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [8,5,3,1,7,6,4,2] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,7,2,3,5,8,6] => [8,5,2,7,6,4,1,3] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,5,2,3,8,4,6,7] => [8,4,7,6,1,5,3,2] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,2,6,7,3,4,8] => [8,4,7,3,2,6,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,2,6,7,8,3,4] => [8,4,7,3,2,1,6,5] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => [8,4,3,7,2,6,5,1] => ? => ? = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000383
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00064: Permutations āreverseā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 67% āvalues known / values provided: 67%ādistinct values known / distinct values provided: 100%
Mp00064: Permutations āreverseā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 67% āvalues known / values provided: 67%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1,2] => [2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,1] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [1,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [2,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [2,1,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [2,1,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => [1,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [1,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [2,1,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => [1,1,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,2,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [2,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,2,1,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [2,1,1,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,2,1,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [2,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,2,1,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [6,8,7,5,4,3,2,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [5,8,7,6,4,3,2,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [4,8,7,6,5,3,2,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,6,3,8,7] => [7,8,3,6,5,4,2,1] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [3,8,7,6,5,4,2,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,2,4,5,7,3,8,6] => [6,8,3,7,5,4,2,1] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => [8,5,3,7,6,4,2,1] => ? => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,4,6,5,8,7] => [7,8,5,6,4,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,2,4,6,8,5,7] => [7,5,8,6,4,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [7,8,4,6,5,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,6,7,4,5,8] => [8,5,4,7,6,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2,6,8,4,5,7] => [7,5,4,8,6,2,3,1] => ? => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,4,6,7,2,5,8] => [8,5,2,7,6,4,3,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,4,5,2,3,7,6,8] => [8,6,7,3,2,5,4,1] => ? => ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => [8,4,3,7,2,6,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,4,7,5,8,6] => [6,8,5,7,4,3,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,4,5,8,7] => [7,8,5,4,6,3,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,4,8,5,6] => [6,5,8,4,7,3,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6,8,7] => [7,8,6,3,5,4,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5,8] => ? => ? => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,3,6,7,8,4] => [4,8,7,6,3,5,1,2] => ? => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,5,7,3,4,6,8] => ? => ? => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => ? = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,6,3,8,7] => [7,8,3,6,5,1,4,2] => ? => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [2,4,5,1,3,7,6,8] => [8,6,7,3,1,5,4,2] => ? => ? = 0 + 1
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => [7,6,3,1,8,5,4,2] => ? => ? = 0 + 1
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,6,1,3,7,8,4,5] => [5,4,8,7,3,1,6,2] => ? => ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,5,1,2,6,7,8] => [8,7,6,2,1,5,4,3] => ? => ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2,8,6,7] => [7,6,8,2,1,5,4,3] => ? => ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
=> [3,5,1,2,4,6,8,7] => [7,8,6,4,2,1,5,3] => ? => ? = 0 + 1
[1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,5,1,6,8,2,4,7] => [7,4,2,8,6,1,5,3] => ? => ? = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [3,5,6,1,2,8,4,7] => [7,4,8,2,1,6,5,3] => ? => ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,2,3,7,6,8] => [8,6,7,3,2,5,1,4] => ? => ? = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
=> [4,1,5,6,2,3,8,7] => [7,8,3,2,6,5,1,4] => ? => ? = 1 + 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [4,1,6,7,8,2,3,5] => [5,3,2,8,7,6,1,4] => ? => ? = 1 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [4,5,7,1,2,3,8,6] => [6,8,3,2,1,7,5,4] => ? => ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => ? = 1 + 1
Description
The last part of an integer composition.
Matching statistic: St000932
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00071: Permutations ādescent compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St000932: Dyck paths ā¶ ā¤Result quality: 49% āvalues known / values provided: 49%ādistinct values known / distinct values provided: 100%
Mp00071: Permutations ādescent compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St000932: Dyck paths ā¶ ā¤Result quality: 49% āvalues known / values provided: 49%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [2] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,4,6,8,5,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,4,7,5,6,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3,5,4,7,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,6,7] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,3,5,6,4,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,5,6,4,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,5,6,7,4,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,3,5,6,8,4,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,3,5,7,4,6,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,4,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,4,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,4,6,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,4,5,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,4,8,5,7] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,4,5] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,4,5,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,4,5,6,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,4,8,5,6] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,4,3,5,7,8,6] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,3,5,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,4,3,6,7,5,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001217
(load all 114 compositions to match this statistic)
(load all 114 compositions to match this statistic)
Mp00027: Dyck paths āto partitionā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001217: Dyck paths ā¶ ā¤Result quality: 42% āvalues known / values provided: 42%ādistinct values known / distinct values provided: 100%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001217: Dyck paths ā¶ ā¤Result quality: 42% āvalues known / values provided: 42%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? = 1
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? = 0
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? = 1
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? = 1
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? = 0
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [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]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? = 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? = 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? = 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,2,1]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000237
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000237: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 100%
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000237: Permutations ā¶ ā¤Result quality: 40% āvalues known / values provided: 40%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [[1,2]]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1,0,0,0]
=> [3,1,2] => [[1,3],[2]]
=> [2,1,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,4,5] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,6,7] => [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,2,4,7,6] => [[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,2,6,7,4] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,7,4,6] => [[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,2,4,5,7] => [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,2,4,7,5] => [[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => ? = 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => [[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => ? = 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,2,4,5,6] => [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,2,3,5,7,6] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,2,3,6,7,5] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,2,5,3,6,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,7,6] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,2,5,7,3,6] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,2,6,3,5,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,7,5] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,6,7,3,5] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,7,3,5,6] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,2,3,6,7] => [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,2,3,7,6] => [[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => ? = 0
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,2,6,7,3] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,2,7,3,6] => [[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,2,7,3] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,4,6,2,3,5,7] => [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => [[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => ? = 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000352The Elizalde-Pak rank of a permutation. St000734The last entry in the first row of a standard tableau. St000990The first ascent of a permutation. St000390The number of runs of ones in a binary word. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001271The competition number of a graph. St000363The number of minimal vertex covers of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000504The cardinality of the first block of a set partition. St000971The smallest closer of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000883The number of longest increasing subsequences of a permutation. St000234The number of global ascents of a permutation. St000007The number of saliances of the permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St000542The number of left-to-right-minima of a permutation. St000068The number of minimal elements in a poset. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000054The first entry of the permutation. St000740The last entry of a permutation. St000989The number of final rises of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000654The first descent of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001498The normalised height of a Nakayama algebra with magnitude 1. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000663The number of right floats of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St001948The number of augmented double ascents of a permutation. St000221The number of strong fixed points of a permutation. St000461The rix statistic of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001330The hat guessing number of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001904The length of the initial strictly increasing segment of a parking function.
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