searching the database
Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000390
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ 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] => 11 => 1
[1,1,0,0]
=> [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 2
[1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 2
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 2
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 2
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000292
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 0 = 1 - 1
[1,0,1,0]
=> [1,1] => 11 => 0 = 1 - 1
[1,1,0,0]
=> [2] => 10 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => 100 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => 100 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 0 = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000291
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 0 => 0 = 1 - 1
[1,0,1,0]
=> [1,1] => 11 => 00 => 0 = 1 - 1
[1,1,0,0]
=> [2] => 10 => 01 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 001 => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 010 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => 100 => 011 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => 100 => 011 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0111 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0111 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0111 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0111 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 00001 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00011 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00011 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 00101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00111 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00111 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00111 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00111 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00111 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 01011 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01011 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 01100 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 01110 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 01111 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 01111 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 01110 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 01111 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 01111 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 01111 => 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,5,4] => 1100001000 => 0011110111 => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,1,4,1] => 1100110001 => 0011001110 => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,1,1,2] => 11011111110 => 00100000001 => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,3] => 10001000100 => 01110111011 => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,5,4] => 1100001000 => 0011110111 => ? = 2 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [3,4,4] => 10010001000 => 01101110111 => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,3,1] => 1100101001 => 0011010110 => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,2,3,1] => 110010101001 => 001101010110 => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,5,4] => 1100001000 => 0011110111 => ? = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000386
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 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]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 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]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 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]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,2] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,2] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,3,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,3,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,5,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,5,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,5,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,5,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000159
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2 - 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> 1
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> 1
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> 1
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> 2
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> 1
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> 1
[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],[]]
=> []
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> 1
[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],[]]
=> []
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000068
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [[1],[]]
=> ([],1)
=> 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]]
=> ([(0,7),(1,6),(1,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
=> ([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]]
=> ([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
=> ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]]
=> ([(0,6),(0,7),(1,3),(1,4),(4,7),(5,2),(6,5)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(5,4)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,3,2] => [[4,3,1,1,1],[2]]
=> ([(0,4),(0,7),(1,3),(1,6),(3,7),(5,2),(6,5)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [[4,3,1,1,1],[2]]
=> ([(0,4),(0,7),(1,3),(1,6),(3,7),(5,2),(6,5)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> ([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
=> ([(0,6),(0,7),(1,3),(1,5),(4,7),(5,4),(6,2)],8)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
=> ([(0,6),(1,4),(1,7),(2,3),(2,6),(3,7),(4,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
=> ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2] => [[4,3,2,1,1],[2,1]]
=> ([(0,6),(0,7),(1,3),(1,6),(2,4),(2,7),(4,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,3,1] => [[4,4,2,1,1],[3,1]]
=> ([(0,7),(1,3),(1,6),(2,4),(2,6),(3,7),(4,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,2,3,1] => [[4,4,2,1,1],[3,1]]
=> ([(0,7),(1,3),(1,6),(2,4),(2,6),(3,7),(4,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> ([(0,3),(0,4),(1,5),(1,6),(4,7),(5,7),(6,2)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,3,3] => [[5,3,1,1],[2]]
=> ([(0,6),(0,7),(1,4),(1,5),(4,7),(5,3),(6,2)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [[5,3,1,1],[2]]
=> ([(0,6),(0,7),(1,4),(1,5),(4,7),(5,3),(6,2)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> ([(0,3),(0,4),(1,5),(1,6),(4,7),(5,7),(6,2)],8)
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
=> ([(0,6),(1,6),(1,7),(2,3),(2,4),(3,7),(4,5)],8)
=> ? = 3
Description
The number of minimal elements in a poset.
Matching statistic: St001037
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[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]
=> 0 = 1 - 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]
=> 0 = 1 - 1
[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 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0 = 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]
=> 0 = 1 - 1
[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 = 2 - 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]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[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 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> 0 = 1 - 1
[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 = 2 - 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]
=> 1 = 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 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 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]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 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
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 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]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 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]
=> ? = 2 - 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
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [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]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[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]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 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]
=> ? = 2 - 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
[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]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [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]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 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]
=> ? = 2 - 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
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[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]
=> ? = 2 - 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
[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]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 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]
=> ? = 3 - 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]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [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]
=> ? = 3 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001085
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [3,5,4,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,5,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,5,2,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,5,2,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => [5,4,7,6,3,2,1] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => [5,7,4,3,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [5,4,7,3,6,2,1] => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [5,4,3,7,6,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => [4,7,3,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [4,3,5,7,6,2,1] => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,4,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,5,6,7,2,1] => [4,3,7,6,5,2,1] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,6,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => [3,7,6,5,4,2,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,7,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => [5,7,4,3,2,6,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,4,6,3,2,7,1] => [5,4,7,3,2,6,1] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => [4,7,6,3,2,5,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => [5,4,3,7,2,6,1] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => [4,7,3,6,2,5,1] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,3,6,5,2,7,1] => [4,3,7,6,2,5,1] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [3,7,6,5,2,4,1] => ? = 2 - 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001124
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? = 1 - 2
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? = 1 - 2
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? = 2 - 2
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? = 1 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? = 1 - 2
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? = 1 - 2
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? = 1 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? = 1 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0 = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0 = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0 = 2 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [[5,4],[3]]
=> [3]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [[5,5],[4]]
=> [4]
=> 0 = 2 - 2
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000356The number of occurrences of the pattern 13-2. St000779The tier of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000092The number of outer peaks of a permutation. St000456The monochromatic index of a connected graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000353The number of inner valleys of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001487The number of inner corners of a skew partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. 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 $eA$. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000768The number of peaks in an integer composition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001728The number of invisible descents of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!