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Your data matches 86 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1] => 1
[1,1,0,0]
=> [2] => 2
[1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [2,1] => 1
[1,1,0,1,0,0]
=> [2,1] => 1
[1,1,1,0,0,0]
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1
[1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1] => [1,1] => 1
[1,1,0,0]
=> [2] => [2] => 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => 1
[1,1,1,0,0,0]
=> [3] => [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => 3
Description
The first part of an integer composition.
Matching statistic: St000326
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => 1 => 1 => 1
[1,0,1,0]
=> [1,1] => 11 => 11 => 1
[1,1,0,0]
=> [2] => 10 => 01 => 2
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 1
[1,1,0,1,0,0]
=> [2,1] => 101 => 101 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 10101 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 10101 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 10101 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 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] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => 1100000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => 1100000011 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,6,2,1] => 1100000101 => 1010000011 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1,1] => 11000000011 => 11000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => 1100000011 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,6,2,1] => 1100000101 => 1010000011 => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,6,1,2] => 1100000110 => 0110000011 => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,5,3,1] => 1100001001 => 1001000011 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,2,6,1] => 1101000001 => 1000001011 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,5,1] => 1111100001 => 1000011111 => ? = 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,3,3,1,1] => 1110010011 => 1100100111 => ? = 1
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,1,1,3,1,1] => 1001110011 => 1100111001 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,5,1] => 1111100001 => 1000011111 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => 1110101111 => 1111010111 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2,2] => 1110101010 => 0101010111 => ? = 2
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,7,1] => 1110000001 => 1000000111 => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,5,1,1,1,1] => 1100001111 => 1111000011 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,7,1,1] => 1100000011 => 1100000011 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => 1011111011 => 1101111101 => ? = 1
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,6,1] => 1011000001 => 1000001101 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => 1010111111 => 1111110101 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,2,1,1,2,2] => 1010111010 => 0101110101 => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,3,2,2] => 1001001010 => 0101001001 => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,4,1,1] => 1000100011 => 1100010001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 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: St000297
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 1
[1,0,1,0]
=> [1,1] => [2] => 10 => 1
[1,1,0,0]
=> [2] => [1,1] => 11 => 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 100 => 1
[1,0,1,1,0,0]
=> [1,2] => [1,2] => 110 => 2
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [2,1] => [2,1] => 101 => 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 111 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 1100 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 1010 => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => 1010 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 1110 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => 1001 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 1101 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => 1001 => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => 1101 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 1011 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => 1011 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => 1011 => 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 11000 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => 10100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => 10100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 11100 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => 10010 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => 10010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => 10010 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => 11010 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 11110 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => 11001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => 10101 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => 10101 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => 11101 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 10001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => 11001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => 10001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => 10001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => 11001 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => 10101 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => 10101 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => 10101 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => 11101 => 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 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] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,6,1,2] => [1,3,1,1,1,1,2] => 1100111110 => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,5,3,1] => [2,1,2,1,1,1,2] => 1011011110 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,2,6,1] => [2,1,1,1,1,2,2] => 1011111010 => ? = 1
[1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [3,1,1,3,1,1] => [3,1,4,1,1] => 1001100011 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => [1,2,7] => 1101000000 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => [5,2,3] => 1000010100 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,4,4] => [1,1,1,2,1,1,3] => 1111011100 => ? = 4
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,7,1] => [2,1,1,1,1,1,3] => 1011111100 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,1,2] => [1,8,1] => 1100000001 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => [3,6,1] => 1001000001 => ? = 1
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,6,1] => [2,1,1,1,1,3,1] => 1011111001 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [7,2,1] => 1000000101 => ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,2,3,3] => [1,1,2,1,2,2,1] => 1110110101 => ? = 3
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,2,3] => [1,1,2,2,2,1,1] => 1110101011 => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,3,2,2] => [1,2,2,1,2,1,1] => 1101011011 => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,1,4] => [1,1,1,4,1,1,1] => 1111000111 => ? = 4
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,4,1,1] => [3,1,1,2,1,1,1] => 1001110111 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,1,1,1,1,2] => [1,8,1] => 1100000001 => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,2,1,1,1,1,1,1] => [7,2,1] => 1000000101 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,2,1,1,1,1,1,1] => [7,2,1] => 1000000101 => ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,2,1,1,1,1] => [5,4,1] => 1000010001 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,1,1,1,1,2,1,1] => [3,6,1] => 1001000001 => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,1,1,2,1,1,1,1] => [5,4,1] => 1000010001 => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,1,1,3,1,1] => [3,1,4,1,1] => 1001100011 => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [3,2,1,2,1,1] => [3,3,2,1,1] => 1001001011 => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000439
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,1,0,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000678
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000010
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => ([],1)
=> [1]
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
[1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St001176
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => ([],1)
=> [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> [2]
=> 0 = 1 - 1
[1,1,0,0]
=> [2] => ([],2)
=> [1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000745
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 90%
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 1
[1,1,0,0]
=> [1,2] => [2,1] => [[1],[2]]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [[1,3],[2]]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => [[1,2],[3]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => [[1,2],[3],[4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,3,2,1] => [1,2,4,5,3,6,7,8] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,7,3,2,1] => [1,4,3,5,2,6,7,8] => ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,7,3,2,1] => [1,3,5,4,2,6,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,3,4,2,1] => [2,3,1,4,6,5,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => [1,3,4,2,6,5,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,3,4,2,1] => [1,4,3,2,6,5,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,4,2,1] => [3,4,2,1,6,5,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,5,2,1] => [2,1,3,5,6,4,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,6,2,1] => [2,1,4,5,6,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,3,8,2,1] => [2,4,3,5,6,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,6,2,1] => [1,2,5,4,6,3,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,6,2,1] => [2,1,5,4,6,3,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => [1,3,5,4,6,2,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,3,8,2,1] => [3,2,5,4,6,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,2,1] => [2,5,4,3,6,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,2,1] => [2,1,3,6,5,4,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => [1,3,4,6,5,2,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,2,1] => [3,2,4,6,5,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => [3,4,2,6,5,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,2,1] => [3,2,5,6,4,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => [4,3,5,6,2,1,7,8] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => [2,5,4,6,3,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,2,1] => [3,5,4,6,2,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,7,8,2,1] => [4,5,3,6,2,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,6,8,2,1] => [2,6,5,4,3,1,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,8,2,1] => [4,6,5,3,2,1,7,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => [2,1,4,3,5,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,4,2,3,1] => [2,4,3,1,5,7,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => [3,4,2,1,5,7,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => [2,1,3,5,4,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => [2,1,4,5,3,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,3,1] => [3,2,4,5,1,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => [4,3,2,5,1,7,6,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,2,3,1] => [1,2,5,4,3,7,6,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,2,3,1] => [4,3,5,2,1,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,3,2,4,1] => [2,1,3,4,6,7,5,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,2,4,1] => [1,3,2,4,6,7,5,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => [2,1,4,3,6,7,5,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => [2,4,3,1,6,7,5,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,2,4,1] => [3,4,2,1,6,7,5,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,2,5,1] => [2,1,3,5,6,7,4,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,2,7,1] => [1,4,3,5,6,7,2,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,3,2,8,1] => [2,4,3,5,6,7,1,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,2,6,1] => [2,1,5,4,6,7,3,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,2,7,1] => [1,3,5,4,6,7,2,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,3,2,8,1] => [3,2,5,4,6,7,1,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => [2,4,5,3,6,7,1,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,2,8,1] => [4,3,5,2,6,7,1,8] => ?
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000759The smallest missing part in an integer partition. St000617The number of global maxima of a Dyck path. St000392The length of the longest run of ones in a binary word. St000288The number of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000883The number of longest increasing subsequences of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001050The number of terminal closers of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000234The number of global ascents of a permutation. St000993The multiplicity of the largest part of an integer partition. St000733The row containing the largest entry of a standard tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000654The first descent of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000717The number of ordinal summands of a poset. St000054The first entry of the permutation. St000765The number of weak records in an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000264The girth of a graph, which is not a tree. St000504The cardinality of the first block of a set partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000286The number of connected components of the complement of a graph. St000335The difference of lower and upper interactions. St000553The number of blocks of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000918The 2-limited packing number of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001545The second Elser number of a connected graph. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000237The number of small exceedances. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000241The number of cyclical small excedances. 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 $eA$. St001330The hat guessing number of a graph. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001937The size of the center of a parking function. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
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