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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000982
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => 1
[2,1] => 0 => 1
[1,2,3] => 11 => 2
[1,3,2] => 10 => 1
[2,1,3] => 01 => 1
[2,3,1] => 00 => 2
[3,1,2] => 00 => 2
[3,2,1] => 00 => 2
[1,2,3,4] => 111 => 3
[1,2,4,3] => 110 => 2
[1,3,2,4] => 101 => 1
[1,3,4,2] => 100 => 2
[1,4,2,3] => 100 => 2
[1,4,3,2] => 100 => 2
[2,1,3,4] => 011 => 2
[2,1,4,3] => 010 => 1
[2,3,1,4] => 001 => 2
[2,3,4,1] => 000 => 3
[2,4,1,3] => 000 => 3
[2,4,3,1] => 000 => 3
[3,1,2,4] => 001 => 2
[3,1,4,2] => 000 => 3
[3,2,1,4] => 001 => 2
[3,2,4,1] => 000 => 3
[3,4,1,2] => 000 => 3
[3,4,2,1] => 000 => 3
[4,1,2,3] => 000 => 3
[4,1,3,2] => 000 => 3
[4,2,1,3] => 000 => 3
[4,2,3,1] => 000 => 3
[4,3,1,2] => 000 => 3
[4,3,2,1] => 000 => 3
[1,2,3,4,5] => 1111 => 4
[1,2,3,5,4] => 1110 => 3
[1,2,4,3,5] => 1101 => 2
[1,2,4,5,3] => 1100 => 2
[1,2,5,3,4] => 1100 => 2
[1,2,5,4,3] => 1100 => 2
[1,3,2,4,5] => 1011 => 2
[1,3,2,5,4] => 1010 => 1
[1,3,4,2,5] => 1001 => 2
[1,3,4,5,2] => 1000 => 3
[1,3,5,2,4] => 1000 => 3
[1,3,5,4,2] => 1000 => 3
[1,4,2,3,5] => 1001 => 2
[1,4,2,5,3] => 1000 => 3
[1,4,3,2,5] => 1001 => 2
[1,4,3,5,2] => 1000 => 3
[1,4,5,2,3] => 1000 => 3
[1,4,5,3,2] => 1000 => 3
Description
The length of the longest constant subword.
Matching statistic: St000147
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => [1]
=> 1
[2,1] => 0 => [1] => [1]
=> 1
[1,2,3] => 11 => [2] => [2]
=> 2
[1,3,2] => 10 => [1,1] => [1,1]
=> 1
[2,1,3] => 01 => [1,1] => [1,1]
=> 1
[2,3,1] => 00 => [2] => [2]
=> 2
[3,1,2] => 00 => [2] => [2]
=> 2
[3,2,1] => 00 => [2] => [2]
=> 2
[1,2,3,4] => 111 => [3] => [3]
=> 3
[1,2,4,3] => 110 => [2,1] => [2,1]
=> 2
[1,3,2,4] => 101 => [1,1,1] => [1,1,1]
=> 1
[1,3,4,2] => 100 => [1,2] => [2,1]
=> 2
[1,4,2,3] => 100 => [1,2] => [2,1]
=> 2
[1,4,3,2] => 100 => [1,2] => [2,1]
=> 2
[2,1,3,4] => 011 => [1,2] => [2,1]
=> 2
[2,1,4,3] => 010 => [1,1,1] => [1,1,1]
=> 1
[2,3,1,4] => 001 => [2,1] => [2,1]
=> 2
[2,3,4,1] => 000 => [3] => [3]
=> 3
[2,4,1,3] => 000 => [3] => [3]
=> 3
[2,4,3,1] => 000 => [3] => [3]
=> 3
[3,1,2,4] => 001 => [2,1] => [2,1]
=> 2
[3,1,4,2] => 000 => [3] => [3]
=> 3
[3,2,1,4] => 001 => [2,1] => [2,1]
=> 2
[3,2,4,1] => 000 => [3] => [3]
=> 3
[3,4,1,2] => 000 => [3] => [3]
=> 3
[3,4,2,1] => 000 => [3] => [3]
=> 3
[4,1,2,3] => 000 => [3] => [3]
=> 3
[4,1,3,2] => 000 => [3] => [3]
=> 3
[4,2,1,3] => 000 => [3] => [3]
=> 3
[4,2,3,1] => 000 => [3] => [3]
=> 3
[4,3,1,2] => 000 => [3] => [3]
=> 3
[4,3,2,1] => 000 => [3] => [3]
=> 3
[1,2,3,4,5] => 1111 => [4] => [4]
=> 4
[1,2,3,5,4] => 1110 => [3,1] => [3,1]
=> 3
[1,2,4,3,5] => 1101 => [2,1,1] => [2,1,1]
=> 2
[1,2,4,5,3] => 1100 => [2,2] => [2,2]
=> 2
[1,2,5,3,4] => 1100 => [2,2] => [2,2]
=> 2
[1,2,5,4,3] => 1100 => [2,2] => [2,2]
=> 2
[1,3,2,4,5] => 1011 => [1,1,2] => [2,1,1]
=> 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,1,1,1]
=> 1
[1,3,4,2,5] => 1001 => [1,2,1] => [2,1,1]
=> 2
[1,3,4,5,2] => 1000 => [1,3] => [3,1]
=> 3
[1,3,5,2,4] => 1000 => [1,3] => [3,1]
=> 3
[1,3,5,4,2] => 1000 => [1,3] => [3,1]
=> 3
[1,4,2,3,5] => 1001 => [1,2,1] => [2,1,1]
=> 2
[1,4,2,5,3] => 1000 => [1,3] => [3,1]
=> 3
[1,4,3,2,5] => 1001 => [1,2,1] => [2,1,1]
=> 2
[1,4,3,5,2] => 1000 => [1,3] => [3,1]
=> 3
[1,4,5,2,3] => 1000 => [1,3] => [3,1]
=> 3
[1,4,5,3,2] => 1000 => [1,3] => [3,1]
=> 3
[2,1,4,3,6,5,10,9,8,7,12,11] => 01010100010 => [1,1,1,1,1,1,3,1,1] => ?
=> ? = 3
[2,1,4,3,8,7,6,5,10,9,12,11] => 01010001010 => [1,1,1,1,3,1,1,1,1] => ?
=> ? = 3
[2,1,4,3,8,7,6,5,12,11,10,9] => 01010001000 => [1,1,1,1,3,1,3] => ?
=> ? = 3
[2,1,6,5,4,3,8,7,10,9,12,11] => 01000101010 => [1,1,3,1,1,1,1,1,1] => ?
=> ? = 3
[2,1,6,5,4,3,8,7,12,11,10,9] => 01000101000 => [1,1,3,1,1,1,3] => ?
=> ? = 3
[2,1,6,5,4,3,10,9,8,7,12,11] => 01000100010 => [1,1,3,1,3,1,1] => ?
=> ? = 3
[2,1,8,5,4,7,6,3,12,11,10,9] => 01000001000 => [1,1,5,1,3] => ?
=> ? = 5
[2,1,8,7,6,5,4,3,12,11,10,9] => 01000001000 => [1,1,5,1,3] => ?
=> ? = 5
[4,3,2,1,6,5,8,7,12,11,10,9] => 00010101000 => [3,1,1,1,1,1,3] => ?
=> ? = 3
[4,3,2,1,6,5,10,9,8,7,12,11] => 00010100010 => [3,1,1,1,3,1,1] => ?
=> ? = 3
[4,3,2,1,8,7,6,5,10,9,12,11] => 00010001010 => [3,1,3,1,1,1,1] => ?
=> ? = 3
[4,3,2,1,8,7,6,5,12,11,10,9] => 00010001000 => [3,1,3,1,3] => ?
=> ? = 3
[4,3,2,1,10,7,6,9,8,5,12,11] => 00010000010 => [3,1,5,1,1] => ?
=> ? = 5
[4,3,2,1,10,9,8,7,6,5,12,11] => 00010000010 => [3,1,5,1,1] => ?
=> ? = 5
[6,3,2,5,4,1,12,9,8,11,10,7] => 00000100000 => [5,1,5] => ?
=> ? = 5
[6,3,2,5,4,1,12,11,10,9,8,7] => 00000100000 => [5,1,5] => ?
=> ? = 5
[10,3,2,5,4,7,6,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,3,2,5,4,9,8,7,6,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,3,2,7,6,5,4,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,3,2,9,6,5,8,7,4,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,3,2,9,8,7,6,5,4,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[6,5,4,3,2,1,12,9,8,11,10,7] => 00000100000 => [5,1,5] => ?
=> ? = 5
[6,5,4,3,2,1,12,11,10,9,8,7] => 00000100000 => [5,1,5] => ?
=> ? = 5
[10,5,4,3,2,7,6,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,5,4,3,2,9,8,7,6,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,7,4,3,6,5,2,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,9,4,3,6,5,8,7,2,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,9,4,3,8,7,6,5,2,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,7,6,5,4,3,2,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,9,6,5,4,3,8,7,2,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,9,8,5,4,7,6,3,2,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,9,8,7,6,5,4,3,2,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[2,3,4,5,6,7,8,9,10,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[2,1,4,3,6,5,9,7,10,8,12,11] => 01010100010 => [1,1,1,1,1,1,3,1,1] => ?
=> ? = 3
[2,1,4,3,7,5,8,6,10,9,12,11] => 01010001010 => [1,1,1,1,3,1,1,1,1] => ?
=> ? = 3
[2,1,4,3,7,5,8,6,11,9,12,10] => 01010001000 => [1,1,1,1,3,1,3] => ?
=> ? = 3
[2,1,5,3,6,4,8,7,10,9,12,11] => 01000101010 => [1,1,3,1,1,1,1,1,1] => ?
=> ? = 3
[2,1,5,3,6,4,8,7,11,9,12,10] => 01000101000 => [1,1,3,1,1,1,3] => ?
=> ? = 3
[2,1,5,3,6,4,9,7,10,8,12,11] => 01000100010 => [1,1,3,1,3,1,1] => ?
=> ? = 3
[2,1,5,3,7,4,8,6,11,9,12,10] => 01000001000 => [1,1,5,1,3] => ?
=> ? = 5
[2,1,6,3,7,4,8,5,11,9,12,10] => 01000001000 => [1,1,5,1,3] => ?
=> ? = 5
[3,1,4,2,6,5,8,7,11,9,12,10] => 00010101000 => [3,1,1,1,1,1,3] => ?
=> ? = 3
[3,1,4,2,6,5,9,7,10,8,12,11] => 00010100010 => [3,1,1,1,3,1,1] => ?
=> ? = 3
[3,1,4,2,7,5,8,6,10,9,12,11] => 00010001010 => [3,1,3,1,1,1,1] => ?
=> ? = 3
[3,1,4,2,7,5,8,6,11,9,12,10] => 00010001000 => [3,1,3,1,3] => ?
=> ? = 3
[3,1,4,2,7,5,9,6,10,8,12,11] => 00010000010 => [3,1,5,1,1] => ?
=> ? = 5
[3,1,4,2,8,5,9,6,10,7,12,11] => 00010000010 => [3,1,5,1,1] => ?
=> ? = 5
[3,1,5,2,6,4,9,7,11,8,12,10] => 00000100000 => [5,1,5] => ?
=> ? = 5
[3,1,5,2,6,4,10,7,11,8,12,9] => 00000100000 => [5,1,5] => ?
=> ? = 5
[3,1,5,2,7,4,9,6,10,8,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
Description
The largest part of an integer partition.
Matching statistic: St000381
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => [1] => 1
[2,1] => 0 => [1] => [1] => 1
[1,2,3] => 11 => [2] => [2] => 2
[1,3,2] => 10 => [1,1] => [1,1] => 1
[2,1,3] => 01 => [1,1] => [1,1] => 1
[2,3,1] => 00 => [2] => [2] => 2
[3,1,2] => 00 => [2] => [2] => 2
[3,2,1] => 00 => [2] => [2] => 2
[1,2,3,4] => 111 => [3] => [3] => 3
[1,2,4,3] => 110 => [2,1] => [1,2] => 2
[1,3,2,4] => 101 => [1,1,1] => [1,1,1] => 1
[1,3,4,2] => 100 => [1,2] => [2,1] => 2
[1,4,2,3] => 100 => [1,2] => [2,1] => 2
[1,4,3,2] => 100 => [1,2] => [2,1] => 2
[2,1,3,4] => 011 => [1,2] => [2,1] => 2
[2,1,4,3] => 010 => [1,1,1] => [1,1,1] => 1
[2,3,1,4] => 001 => [2,1] => [1,2] => 2
[2,3,4,1] => 000 => [3] => [3] => 3
[2,4,1,3] => 000 => [3] => [3] => 3
[2,4,3,1] => 000 => [3] => [3] => 3
[3,1,2,4] => 001 => [2,1] => [1,2] => 2
[3,1,4,2] => 000 => [3] => [3] => 3
[3,2,1,4] => 001 => [2,1] => [1,2] => 2
[3,2,4,1] => 000 => [3] => [3] => 3
[3,4,1,2] => 000 => [3] => [3] => 3
[3,4,2,1] => 000 => [3] => [3] => 3
[4,1,2,3] => 000 => [3] => [3] => 3
[4,1,3,2] => 000 => [3] => [3] => 3
[4,2,1,3] => 000 => [3] => [3] => 3
[4,2,3,1] => 000 => [3] => [3] => 3
[4,3,1,2] => 000 => [3] => [3] => 3
[4,3,2,1] => 000 => [3] => [3] => 3
[1,2,3,4,5] => 1111 => [4] => [4] => 4
[1,2,3,5,4] => 1110 => [3,1] => [1,3] => 3
[1,2,4,3,5] => 1101 => [2,1,1] => [1,2,1] => 2
[1,2,4,5,3] => 1100 => [2,2] => [2,2] => 2
[1,2,5,3,4] => 1100 => [2,2] => [2,2] => 2
[1,2,5,4,3] => 1100 => [2,2] => [2,2] => 2
[1,3,2,4,5] => 1011 => [1,1,2] => [2,1,1] => 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,1,1,1] => 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,1,2] => 2
[1,3,4,5,2] => 1000 => [1,3] => [3,1] => 3
[1,3,5,2,4] => 1000 => [1,3] => [3,1] => 3
[1,3,5,4,2] => 1000 => [1,3] => [3,1] => 3
[1,4,2,3,5] => 1001 => [1,2,1] => [1,1,2] => 2
[1,4,2,5,3] => 1000 => [1,3] => [3,1] => 3
[1,4,3,2,5] => 1001 => [1,2,1] => [1,1,2] => 2
[1,4,3,5,2] => 1000 => [1,3] => [3,1] => 3
[1,4,5,2,3] => 1000 => [1,3] => [3,1] => 3
[1,4,5,3,2] => 1000 => [1,3] => [3,1] => 3
[2,1,4,3,6,5,8,7,10,9,12,11] => 01010101010 => [1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => ? = 1
[2,1,4,3,6,5,10,9,8,7,12,11] => 01010100010 => [1,1,1,1,1,1,3,1,1] => ? => ? = 3
[2,1,4,3,8,7,6,5,10,9,12,11] => 01010001010 => [1,1,1,1,3,1,1,1,1] => ? => ? = 3
[2,1,4,3,8,7,6,5,12,11,10,9] => 01010001000 => [1,1,1,1,3,1,3] => ? => ? = 3
[2,1,6,5,4,3,8,7,10,9,12,11] => 01000101010 => [1,1,3,1,1,1,1,1,1] => ? => ? = 3
[2,1,6,5,4,3,8,7,12,11,10,9] => 01000101000 => [1,1,3,1,1,1,3] => ? => ? = 3
[2,1,6,5,4,3,10,9,8,7,12,11] => 01000100010 => [1,1,3,1,3,1,1] => ? => ? = 3
[2,1,8,5,4,7,6,3,12,11,10,9] => 01000001000 => [1,1,5,1,3] => ? => ? = 5
[2,1,12,5,4,7,6,9,8,11,10,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,5,4,7,6,11,10,9,8,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,5,4,9,8,7,6,11,10,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,5,4,11,8,7,10,9,6,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,5,4,11,10,9,8,7,6,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,8,7,6,5,4,3,12,11,10,9] => 01000001000 => [1,1,5,1,3] => ? => ? = 5
[2,1,12,7,6,5,4,9,8,11,10,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,7,6,5,4,11,10,9,8,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,9,6,5,8,7,4,11,10,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,11,6,5,8,7,10,9,4,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,11,6,5,10,9,8,7,4,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,9,8,7,6,5,4,11,10,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,11,8,7,6,5,10,9,4,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,11,10,7,6,9,8,5,4,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[2,1,12,11,10,9,8,7,6,5,4,3] => 01000000000 => [1,1,9] => [9,1,1] => ? = 9
[4,3,2,1,6,5,8,7,12,11,10,9] => 00010101000 => [3,1,1,1,1,1,3] => ? => ? = 3
[4,3,2,1,6,5,10,9,8,7,12,11] => 00010100010 => [3,1,1,1,3,1,1] => ? => ? = 3
[4,3,2,1,8,7,6,5,10,9,12,11] => 00010001010 => [3,1,3,1,1,1,1] => ? => ? = 3
[4,3,2,1,8,7,6,5,12,11,10,9] => 00010001000 => [3,1,3,1,3] => ? => ? = 3
[4,3,2,1,10,7,6,9,8,5,12,11] => 00010000010 => [3,1,5,1,1] => ? => ? = 5
[4,3,2,1,10,9,8,7,6,5,12,11] => 00010000010 => [3,1,5,1,1] => ? => ? = 5
[6,3,2,5,4,1,12,9,8,11,10,7] => 00000100000 => [5,1,5] => ? => ? = 5
[6,3,2,5,4,1,12,11,10,9,8,7] => 00000100000 => [5,1,5] => ? => ? = 5
[10,3,2,5,4,7,6,9,8,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,3,2,5,4,9,8,7,6,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,3,2,7,6,5,4,9,8,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,3,2,9,6,5,8,7,4,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,3,2,9,8,7,6,5,4,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[6,5,4,3,2,1,12,9,8,11,10,7] => 00000100000 => [5,1,5] => ? => ? = 5
[6,5,4,3,2,1,12,11,10,9,8,7] => 00000100000 => [5,1,5] => ? => ? = 5
[10,5,4,3,2,7,6,9,8,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,5,4,3,2,9,8,7,6,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,7,4,3,6,5,2,9,8,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,9,4,3,6,5,8,7,2,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,9,4,3,8,7,6,5,2,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,7,6,5,4,3,2,9,8,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,9,6,5,4,3,8,7,2,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,9,8,5,4,7,6,3,2,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[10,9,8,7,6,5,4,3,2,1,12,11] => 00000000010 => [9,1,1] => ? => ? = 9
[11,1,2,3,4,5,6,7,8,9,10] => 0000000000 => [10] => [10] => ? = 10
[2,3,4,5,6,7,8,9,10,11,1] => 0000000000 => [10] => [10] => ? = 10
[2,3,4,5,6,7,8,9,11,1,10] => 0000000000 => [10] => [10] => ? = 10
Description
The largest part of an integer composition.
Matching statistic: St000013
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => [1,0]
=> 1
[2,1] => 0 => [1] => [1,0]
=> 1
[1,2,3] => 11 => [2] => [1,1,0,0]
=> 2
[1,3,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[2,1,3] => 01 => [1,1] => [1,0,1,0]
=> 1
[2,3,1] => 00 => [2] => [1,1,0,0]
=> 2
[3,1,2] => 00 => [2] => [1,1,0,0]
=> 2
[3,2,1] => 00 => [2] => [1,1,0,0]
=> 2
[1,2,3,4] => 111 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,4,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,4,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[2,4,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[2,4,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,1,4,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,4,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,3,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,6,5,4,3,8,7,10,9] => 010001010 => [1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[2,1,8,5,4,7,6,3,10,9] => 010000010 => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[2,1,4,3,8,7,6,5,10,9] => 010100010 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[4,3,2,1,8,7,6,5,10,9] => 000100010 => [3,1,3,1,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[2,1,8,7,6,5,4,3,10,9] => 010000010 => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[2,1,4,3,10,7,6,9,8,5] => 010100000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[4,3,2,1,10,7,6,9,8,5] => 000100000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[4,3,2,1,6,5,10,9,8,7] => 000101000 => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[6,3,2,5,4,1,10,9,8,7] => 000001000 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[2,1,6,5,4,3,10,9,8,7] => 010001000 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[6,5,4,3,2,1,10,9,8,7] => 000001000 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[2,1,4,3,10,9,8,7,6,5] => 010100000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[4,3,2,1,10,9,8,7,6,5] => 000100000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[2,3,4,5,6,1,9,7,8] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[2,3,4,5,1,8,9,6,7] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[2,3,4,1,7,8,9,5,6] => 00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[2,1,4,3,6,5,8,7,10,9,12,11] => 01010101010 => [1,1,1,1,1,1,1,1,1,1,1] => ?
=> ? = 1
[2,1,4,3,6,5,10,9,8,7,12,11] => 01010100010 => [1,1,1,1,1,1,3,1,1] => ?
=> ? = 3
[2,1,4,3,8,7,6,5,10,9,12,11] => 01010001010 => [1,1,1,1,3,1,1,1,1] => ?
=> ? = 3
[2,1,4,3,8,7,6,5,12,11,10,9] => 01010001000 => [1,1,1,1,3,1,3] => ?
=> ? = 3
[2,1,6,5,4,3,8,7,10,9,12,11] => 01000101010 => [1,1,3,1,1,1,1,1,1] => ?
=> ? = 3
[2,1,6,5,4,3,8,7,12,11,10,9] => 01000101000 => [1,1,3,1,1,1,3] => ?
=> ? = 3
[2,1,6,5,4,3,10,9,8,7,12,11] => 01000100010 => [1,1,3,1,3,1,1] => ?
=> ? = 3
[2,1,8,5,4,7,6,3,12,11,10,9] => 01000001000 => [1,1,5,1,3] => ?
=> ? = 5
[2,1,12,5,4,7,6,9,8,11,10,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,5,4,7,6,11,10,9,8,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,5,4,9,8,7,6,11,10,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,5,4,11,8,7,10,9,6,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,5,4,11,10,9,8,7,6,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,8,7,6,5,4,3,12,11,10,9] => 01000001000 => [1,1,5,1,3] => ?
=> ? = 5
[2,1,12,7,6,5,4,9,8,11,10,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,7,6,5,4,11,10,9,8,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,9,6,5,8,7,4,11,10,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,11,6,5,8,7,10,9,4,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,11,6,5,10,9,8,7,4,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,9,8,7,6,5,4,11,10,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,11,8,7,6,5,10,9,4,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,11,10,7,6,9,8,5,4,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[2,1,12,11,10,9,8,7,6,5,4,3] => 01000000000 => [1,1,9] => ?
=> ? = 9
[4,3,2,1,6,5,8,7,12,11,10,9] => 00010101000 => [3,1,1,1,1,1,3] => ?
=> ? = 3
[4,3,2,1,6,5,10,9,8,7,12,11] => 00010100010 => [3,1,1,1,3,1,1] => ?
=> ? = 3
[4,3,2,1,8,7,6,5,10,9,12,11] => 00010001010 => [3,1,3,1,1,1,1] => ?
=> ? = 3
[4,3,2,1,8,7,6,5,12,11,10,9] => 00010001000 => [3,1,3,1,3] => ?
=> ? = 3
[4,3,2,1,10,7,6,9,8,5,12,11] => 00010000010 => [3,1,5,1,1] => ?
=> ? = 5
[4,3,2,1,10,9,8,7,6,5,12,11] => 00010000010 => [3,1,5,1,1] => ?
=> ? = 5
[6,3,2,5,4,1,12,9,8,11,10,7] => 00000100000 => [5,1,5] => ?
=> ? = 5
[6,3,2,5,4,1,12,11,10,9,8,7] => 00000100000 => [5,1,5] => ?
=> ? = 5
[10,3,2,5,4,7,6,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,3,2,5,4,9,8,7,6,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
[10,3,2,7,6,5,4,9,8,1,12,11] => 00000000010 => [9,1,1] => ?
=> ? = 9
Description
The height of a Dyck path.
The height of a Dyck path D of semilength n is defined as the maximal height of a peak of D. The height of D at position i is the number of up-steps minus the number of down-steps before position i.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 99%●distinct values known / distinct values provided: 90%
Mp00105: Binary words —complement⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 99%●distinct values known / distinct values provided: 90%
Values
[1,2] => 1 => 0 => 0 => 1
[2,1] => 0 => 1 => 1 => 1
[1,2,3] => 11 => 00 => 01 => 2
[1,3,2] => 10 => 01 => 00 => 1
[2,1,3] => 01 => 10 => 11 => 1
[2,3,1] => 00 => 11 => 10 => 2
[3,1,2] => 00 => 11 => 10 => 2
[3,2,1] => 00 => 11 => 10 => 2
[1,2,3,4] => 111 => 000 => 010 => 3
[1,2,4,3] => 110 => 001 => 011 => 2
[1,3,2,4] => 101 => 010 => 000 => 1
[1,3,4,2] => 100 => 011 => 001 => 2
[1,4,2,3] => 100 => 011 => 001 => 2
[1,4,3,2] => 100 => 011 => 001 => 2
[2,1,3,4] => 011 => 100 => 110 => 2
[2,1,4,3] => 010 => 101 => 111 => 1
[2,3,1,4] => 001 => 110 => 100 => 2
[2,3,4,1] => 000 => 111 => 101 => 3
[2,4,1,3] => 000 => 111 => 101 => 3
[2,4,3,1] => 000 => 111 => 101 => 3
[3,1,2,4] => 001 => 110 => 100 => 2
[3,1,4,2] => 000 => 111 => 101 => 3
[3,2,1,4] => 001 => 110 => 100 => 2
[3,2,4,1] => 000 => 111 => 101 => 3
[3,4,1,2] => 000 => 111 => 101 => 3
[3,4,2,1] => 000 => 111 => 101 => 3
[4,1,2,3] => 000 => 111 => 101 => 3
[4,1,3,2] => 000 => 111 => 101 => 3
[4,2,1,3] => 000 => 111 => 101 => 3
[4,2,3,1] => 000 => 111 => 101 => 3
[4,3,1,2] => 000 => 111 => 101 => 3
[4,3,2,1] => 000 => 111 => 101 => 3
[1,2,3,4,5] => 1111 => 0000 => 0101 => 4
[1,2,3,5,4] => 1110 => 0001 => 0100 => 3
[1,2,4,3,5] => 1101 => 0010 => 0111 => 2
[1,2,4,5,3] => 1100 => 0011 => 0110 => 2
[1,2,5,3,4] => 1100 => 0011 => 0110 => 2
[1,2,5,4,3] => 1100 => 0011 => 0110 => 2
[1,3,2,4,5] => 1011 => 0100 => 0001 => 2
[1,3,2,5,4] => 1010 => 0101 => 0000 => 1
[1,3,4,2,5] => 1001 => 0110 => 0011 => 2
[1,3,4,5,2] => 1000 => 0111 => 0010 => 3
[1,3,5,2,4] => 1000 => 0111 => 0010 => 3
[1,3,5,4,2] => 1000 => 0111 => 0010 => 3
[1,4,2,3,5] => 1001 => 0110 => 0011 => 2
[1,4,2,5,3] => 1000 => 0111 => 0010 => 3
[1,4,3,2,5] => 1001 => 0110 => 0011 => 2
[1,4,3,5,2] => 1000 => 0111 => 0010 => 3
[1,4,5,2,3] => 1000 => 0111 => 0010 => 3
[1,4,5,3,2] => 1000 => 0111 => 0010 => 3
[2,1,4,3,6,5,8,7,10,9,12,11] => 01010101010 => 10101010101 => 11111111111 => ? = 1
[2,1,4,3,6,5,10,9,8,7,12,11] => 01010100010 => 10101011101 => 11111110111 => ? = 3
[2,1,4,3,8,7,6,5,10,9,12,11] => 01010001010 => 10101110101 => 11111011111 => ? = 3
[2,1,4,3,8,7,6,5,12,11,10,9] => 01010001000 => 10101110111 => 11111011101 => ? = 3
[2,1,6,5,4,3,8,7,10,9,12,11] => 01000101010 => 10111010101 => 11101111111 => ? = 3
[2,1,6,5,4,3,8,7,12,11,10,9] => 01000101000 => 10111010111 => 11101111101 => ? = 3
[2,1,6,5,4,3,10,9,8,7,12,11] => 01000100010 => 10111011101 => 11101110111 => ? = 3
[2,1,8,5,4,7,6,3,12,11,10,9] => 01000001000 => 10111110111 => 11101011101 => ? = 5
[2,1,12,5,4,7,6,9,8,11,10,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,5,4,7,6,11,10,9,8,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,5,4,9,8,7,6,11,10,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,5,4,11,8,7,10,9,6,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,5,4,11,10,9,8,7,6,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,8,7,6,5,4,3,12,11,10,9] => 01000001000 => 10111110111 => 11101011101 => ? = 5
[2,1,12,7,6,5,4,9,8,11,10,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,7,6,5,4,11,10,9,8,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,9,6,5,8,7,4,11,10,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,11,6,5,8,7,10,9,4,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,11,6,5,10,9,8,7,4,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,9,8,7,6,5,4,11,10,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,11,8,7,6,5,10,9,4,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,11,10,7,6,9,8,5,4,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[2,1,12,11,10,9,8,7,6,5,4,3] => 01000000000 => 10111111111 => 11101010101 => ? = 9
[4,3,2,1,6,5,8,7,12,11,10,9] => 00010101000 => 11101010111 => 10111111101 => ? = 3
[4,3,2,1,6,5,10,9,8,7,12,11] => 00010100010 => 11101011101 => 10111110111 => ? = 3
[4,3,2,1,8,7,6,5,10,9,12,11] => 00010001010 => 11101110101 => 10111011111 => ? = 3
[4,3,2,1,8,7,6,5,12,11,10,9] => 00010001000 => 11101110111 => 10111011101 => ? = 3
[4,3,2,1,10,7,6,9,8,5,12,11] => 00010000010 => 11101111101 => 10111010111 => ? = 5
[4,3,2,1,10,9,8,7,6,5,12,11] => 00010000010 => 11101111101 => 10111010111 => ? = 5
[6,3,2,5,4,1,12,9,8,11,10,7] => 00000100000 => 11111011111 => 10101110101 => ? = 5
[6,3,2,5,4,1,12,11,10,9,8,7] => 00000100000 => 11111011111 => 10101110101 => ? = 5
[10,3,2,5,4,7,6,9,8,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,3,2,5,4,9,8,7,6,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,3,2,7,6,5,4,9,8,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,3,2,9,6,5,8,7,4,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,3,2,9,8,7,6,5,4,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[6,5,4,3,2,1,12,9,8,11,10,7] => 00000100000 => 11111011111 => 10101110101 => ? = 5
[6,5,4,3,2,1,12,11,10,9,8,7] => 00000100000 => 11111011111 => 10101110101 => ? = 5
[10,5,4,3,2,7,6,9,8,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,5,4,3,2,9,8,7,6,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,7,4,3,6,5,2,9,8,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,9,4,3,6,5,8,7,2,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,9,4,3,8,7,6,5,2,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,7,6,5,4,3,2,9,8,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,9,6,5,4,3,8,7,2,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,9,8,5,4,7,6,3,2,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[10,9,8,7,6,5,4,3,2,1,12,11] => 00000000010 => 11111111101 => 10101010111 => ? = 9
[1,2,3,4,5,6,7,8,9,11,10] => 1111111110 => 0000000001 => 0101010100 => ? = 9
[2,1,3,4,5,6,7,8,9,10,11] => 0111111111 => 1000000000 => 1101010101 => ? = 9
[11,1,2,3,4,5,6,7,8,9,10] => 0000000000 => 1111111111 => 1010101010 => ? = 10
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form 010... or of the form 101....
Matching statistic: St000444
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 89%●distinct values known / distinct values provided: 70%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 89%●distinct values known / distinct values provided: 70%
Values
[1,2] => 1 => [1] => [1,0]
=> ? = 1
[2,1] => 0 => [1] => [1,0]
=> ? = 1
[1,2,3] => 11 => [2] => [1,1,0,0]
=> 2
[1,3,2] => 10 => [1,1] => [1,0,1,0]
=> 1
[2,1,3] => 01 => [1,1] => [1,0,1,0]
=> 1
[2,3,1] => 00 => [2] => [1,1,0,0]
=> 2
[3,1,2] => 00 => [2] => [1,1,0,0]
=> 2
[3,2,1] => 00 => [2] => [1,1,0,0]
=> 2
[1,2,3,4] => 111 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,4,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,4,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[2,4,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[2,4,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,1,4,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[3,4,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[4,3,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,5,2,3,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,5,2,4,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,4,3,6,5,8,7,10,9] => 010101010 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[3,4,1,2,6,5,8,7,10,9] => 000101010 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,3,2,1,6,5,8,7,10,9] => 000101010 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[6,3,2,5,4,1,8,7,10,9] => 000001010 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[8,3,2,5,4,7,6,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7
[10,3,2,5,4,7,6,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,6,5,4,3,8,7,10,9] => 010001010 => [1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[6,5,4,3,2,1,8,7,10,9] => 000001010 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[8,5,4,3,2,7,6,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7
[10,5,4,3,2,7,6,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,8,5,4,7,6,3,10,9] => 010000010 => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[8,7,4,3,6,5,2,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7
[10,7,4,3,6,5,2,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,10,5,4,7,6,9,8,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[10,9,4,3,6,5,8,7,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[8,9,5,6,3,4,10,1,2,7] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,4,3,8,7,6,5,10,9] => 010100010 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[4,3,2,1,8,7,6,5,10,9] => 000100010 => [3,1,3,1,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[8,3,2,7,6,5,4,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7
[10,3,2,7,6,5,4,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,8,7,6,5,4,3,10,9] => 010000010 => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[8,7,6,5,4,3,2,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7
[9,7,6,5,4,3,2,10,1,8] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[10,7,6,5,4,3,2,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,10,7,6,5,4,9,8,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[10,9,6,5,4,3,8,7,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,4,3,10,7,6,9,8,5] => 010100000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[4,3,2,1,10,7,6,9,8,5] => 000100000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[10,3,2,9,6,5,8,7,4,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,10,9,6,5,8,7,4,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[10,9,8,5,4,7,6,3,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[10,8,9,6,7,4,5,2,3,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[6,7,8,9,10,1,2,3,4,5] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[7,3,2,8,9,10,1,4,5,6] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[8,5,4,3,2,9,10,1,6,7] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,4,3,6,5,10,9,8,7] => 010101000 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[4,3,2,1,6,5,10,9,8,7] => 000101000 => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[6,3,2,5,4,1,10,9,8,7] => 000001000 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[10,3,2,5,4,9,8,7,6,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,6,5,4,3,10,9,8,7] => 010001000 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[6,5,4,3,2,1,10,9,8,7] => 000001000 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[10,5,4,3,2,9,8,7,6,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,10,5,4,9,8,7,6,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[10,9,4,3,8,7,6,5,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[9,7,5,10,3,8,2,6,1,4] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[2,1,4,3,10,9,8,7,6,5] => 010100000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[4,3,2,1,10,9,8,7,6,5] => 000100000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[10,3,2,9,8,7,6,5,4,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 89%●distinct values known / distinct values provided: 70%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 89%●distinct values known / distinct values provided: 70%
Values
[1,2] => 1 => [1] => [1,0]
=> ? = 1 - 1
[2,1] => 0 => [1] => [1,0]
=> ? = 1 - 1
[1,2,3] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => 10 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1,3] => 01 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,3,1] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[3,1,2] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => 00 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3,4] => 111 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[2,4,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[2,4,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,4,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,4,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,4,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,4,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[4,1,2,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[4,1,3,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[4,2,1,3] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[4,2,3,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[4,3,1,2] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[4,3,2,1] => 000 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,3,4,5] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,5,2,3,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,5,2,4,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,4,3,6,5,8,7,10,9] => 010101010 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[3,4,1,2,6,5,8,7,10,9] => 000101010 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[4,3,2,1,6,5,8,7,10,9] => 000101010 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,3,2,5,4,1,8,7,10,9] => 000001010 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[8,3,2,5,4,7,6,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[10,3,2,5,4,7,6,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,6,5,4,3,8,7,10,9] => 010001010 => [1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,5,4,3,2,1,8,7,10,9] => 000001010 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[8,5,4,3,2,7,6,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[10,5,4,3,2,7,6,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,8,5,4,7,6,3,10,9] => 010000010 => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[8,7,4,3,6,5,2,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[10,7,4,3,6,5,2,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,10,5,4,7,6,9,8,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[10,9,4,3,6,5,8,7,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,9,5,6,3,4,10,1,2,7] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,4,3,8,7,6,5,10,9] => 010100010 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[4,3,2,1,8,7,6,5,10,9] => 000100010 => [3,1,3,1,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[8,3,2,7,6,5,4,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[10,3,2,7,6,5,4,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,8,7,6,5,4,3,10,9] => 010000010 => [1,1,5,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[8,7,6,5,4,3,2,1,10,9] => 000000010 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[9,7,6,5,4,3,2,10,1,8] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[10,7,6,5,4,3,2,9,8,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,10,7,6,5,4,9,8,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[10,9,6,5,4,3,8,7,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,4,3,10,7,6,9,8,5] => 010100000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[4,3,2,1,10,7,6,9,8,5] => 000100000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[10,3,2,9,6,5,8,7,4,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,10,9,6,5,8,7,4,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[10,9,8,5,4,7,6,3,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[10,8,9,6,7,4,5,2,3,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,8,9,10,1,2,3,4,5] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,3,2,8,9,10,1,4,5,6] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,5,4,3,2,9,10,1,6,7] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,4,3,6,5,10,9,8,7] => 010101000 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[4,3,2,1,6,5,10,9,8,7] => 000101000 => [3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[6,3,2,5,4,1,10,9,8,7] => 000001000 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[10,3,2,5,4,9,8,7,6,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,6,5,4,3,10,9,8,7] => 010001000 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[6,5,4,3,2,1,10,9,8,7] => 000001000 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[10,5,4,3,2,9,8,7,6,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,10,5,4,9,8,7,6,3] => 010000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[10,9,4,3,8,7,6,5,2,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[9,7,5,10,3,8,2,6,1,4] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,4,3,10,9,8,7,6,5] => 010100000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[4,3,2,1,10,9,8,7,6,5] => 000100000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[10,3,2,9,8,7,6,5,4,1] => 000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001235
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 60%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 60%
Values
[1,2] => 1 => [1] => [1] => 1
[2,1] => 0 => [1] => [1] => 1
[1,2,3] => 11 => [2] => [1,1] => 2
[1,3,2] => 10 => [1,1] => [2] => 1
[2,1,3] => 01 => [1,1] => [2] => 1
[2,3,1] => 00 => [2] => [1,1] => 2
[3,1,2] => 00 => [2] => [1,1] => 2
[3,2,1] => 00 => [2] => [1,1] => 2
[1,2,3,4] => 111 => [3] => [1,1,1] => 3
[1,2,4,3] => 110 => [2,1] => [1,2] => 2
[1,3,2,4] => 101 => [1,1,1] => [3] => 1
[1,3,4,2] => 100 => [1,2] => [2,1] => 2
[1,4,2,3] => 100 => [1,2] => [2,1] => 2
[1,4,3,2] => 100 => [1,2] => [2,1] => 2
[2,1,3,4] => 011 => [1,2] => [2,1] => 2
[2,1,4,3] => 010 => [1,1,1] => [3] => 1
[2,3,1,4] => 001 => [2,1] => [1,2] => 2
[2,3,4,1] => 000 => [3] => [1,1,1] => 3
[2,4,1,3] => 000 => [3] => [1,1,1] => 3
[2,4,3,1] => 000 => [3] => [1,1,1] => 3
[3,1,2,4] => 001 => [2,1] => [1,2] => 2
[3,1,4,2] => 000 => [3] => [1,1,1] => 3
[3,2,1,4] => 001 => [2,1] => [1,2] => 2
[3,2,4,1] => 000 => [3] => [1,1,1] => 3
[3,4,1,2] => 000 => [3] => [1,1,1] => 3
[3,4,2,1] => 000 => [3] => [1,1,1] => 3
[4,1,2,3] => 000 => [3] => [1,1,1] => 3
[4,1,3,2] => 000 => [3] => [1,1,1] => 3
[4,2,1,3] => 000 => [3] => [1,1,1] => 3
[4,2,3,1] => 000 => [3] => [1,1,1] => 3
[4,3,1,2] => 000 => [3] => [1,1,1] => 3
[4,3,2,1] => 000 => [3] => [1,1,1] => 3
[1,2,3,4,5] => 1111 => [4] => [1,1,1,1] => 4
[1,2,3,5,4] => 1110 => [3,1] => [1,1,2] => 3
[1,2,4,3,5] => 1101 => [2,1,1] => [1,3] => 2
[1,2,4,5,3] => 1100 => [2,2] => [1,2,1] => 2
[1,2,5,3,4] => 1100 => [2,2] => [1,2,1] => 2
[1,2,5,4,3] => 1100 => [2,2] => [1,2,1] => 2
[1,3,2,4,5] => 1011 => [1,1,2] => [3,1] => 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [4] => 1
[1,3,4,2,5] => 1001 => [1,2,1] => [2,2] => 2
[1,3,4,5,2] => 1000 => [1,3] => [2,1,1] => 3
[1,3,5,2,4] => 1000 => [1,3] => [2,1,1] => 3
[1,3,5,4,2] => 1000 => [1,3] => [2,1,1] => 3
[1,4,2,3,5] => 1001 => [1,2,1] => [2,2] => 2
[1,4,2,5,3] => 1000 => [1,3] => [2,1,1] => 3
[1,4,3,2,5] => 1001 => [1,2,1] => [2,2] => 2
[1,4,3,5,2] => 1000 => [1,3] => [2,1,1] => 3
[1,4,5,2,3] => 1000 => [1,3] => [2,1,1] => 3
[1,4,5,3,2] => 1000 => [1,3] => [2,1,1] => 3
[8,7,6,5,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,8,6,5,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,8,5,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,7,8,5,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,5,6,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,8,5,6,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,6,5,7,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,5,6,7,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,5,8,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,7,5,8,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,5,6,8,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,5,7,8,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[5,6,7,8,4,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,6,4,5,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,8,6,4,5,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,6,7,4,5,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,5,4,6,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,4,5,6,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,5,4,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,7,5,4,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,5,6,4,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,5,7,4,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[5,6,7,4,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,4,5,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,7,4,5,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,5,4,6,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,4,5,6,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,5,4,7,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[5,6,4,7,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,4,5,7,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[5,4,6,7,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[4,5,6,7,8,3,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,6,5,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,8,6,5,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,6,7,5,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,8,5,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,5,6,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,8,5,6,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,5,6,7,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,5,8,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,7,5,8,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,5,7,8,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[5,6,7,8,3,4,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,8,4,3,5,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,6,3,4,5,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,8,6,3,4,5,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,6,7,3,4,5,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[7,6,8,3,4,5,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[6,7,8,3,4,5,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
[8,7,5,4,3,6,2,1] => 0000000 => [7] => [1,1,1,1,1,1,1] => ? = 7
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001645
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 60%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 1 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 2 + 1
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 1 + 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 3 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,4,5,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,5,1,3,4] => [3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,5,1,4,3] => [3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,5,3,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,5,4,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,1,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,2,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,4,2,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,1,2,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,2,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,1,2,5,3] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,1,3,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,1,5,2,3] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,1,5,3,2] => [5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 60%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 2
[1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 1
[2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 2
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[2,4,1,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3
[3,4,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[3,4,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,2,1,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,3,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,5,3,2,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,5,3,4,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,5,4,2,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,5,4,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1,2,4,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1,3,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1,3,4,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1,4,2,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,1,4,3,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,2,1,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,2,1,4,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,2,3,1,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,2,3,4,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,2,4,1,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,2,4,3,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,3,1,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,3,1,4,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,3,2,1,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,3,2,4,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,3,4,1,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,3,4,2,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,4,1,2,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,4,1,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,4,2,1,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,4,2,3,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,4,3,1,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5,4,3,2,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,2,3,5,4] => [1,6,4,3,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,2,4,3,5] => [1,6,5,3,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,2,4,5,3] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,2,5,4,3] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,3,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,3,2,5,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,3,4,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,3,4,5,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,3,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,3,5,4,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,4,2,5,3] => [1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,4,3,2,5] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,1,4,3,5,2] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset.
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