Your data matches 65 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00095: Integer partitions to binary wordBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 1
[2]
=> 100 => 1
[1,1]
=> 110 => 2
[3]
=> 1000 => 1
[2,1]
=> 1010 => 1
[1,1,1]
=> 1110 => 3
[4]
=> 10000 => 1
[3,1]
=> 10010 => 1
[2,2]
=> 1100 => 2
[2,1,1]
=> 10110 => 1
[1,1,1,1]
=> 11110 => 4
[5]
=> 100000 => 1
[4,1]
=> 100010 => 1
[3,2]
=> 10100 => 1
[3,1,1]
=> 100110 => 1
[2,2,1]
=> 11010 => 2
[2,1,1,1]
=> 101110 => 1
[1,1,1,1,1]
=> 111110 => 5
[6]
=> 1000000 => 1
[5,1]
=> 1000010 => 1
[4,2]
=> 100100 => 1
[4,1,1]
=> 1000110 => 1
[3,3]
=> 11000 => 2
[3,2,1]
=> 101010 => 1
[3,1,1,1]
=> 1001110 => 1
[2,2,2]
=> 11100 => 3
[2,2,1,1]
=> 110110 => 2
[2,1,1,1,1]
=> 1011110 => 1
[1,1,1,1,1,1]
=> 1111110 => 6
[7]
=> 10000000 => 1
[6,1]
=> 10000010 => 1
[5,2]
=> 1000100 => 1
[5,1,1]
=> 10000110 => 1
[4,3]
=> 101000 => 1
[4,2,1]
=> 1001010 => 1
[4,1,1,1]
=> 10001110 => 1
[3,3,1]
=> 110010 => 2
[3,2,2]
=> 101100 => 1
[3,2,1,1]
=> 1010110 => 1
[3,1,1,1,1]
=> 10011110 => 1
[2,2,2,1]
=> 111010 => 3
[2,2,1,1,1]
=> 1101110 => 2
[2,1,1,1,1,1]
=> 10111110 => 1
[1,1,1,1,1,1,1]
=> 11111110 => 7
[8]
=> 100000000 => 1
[7,1]
=> 100000010 => 1
[6,2]
=> 10000100 => 1
[6,1,1]
=> 100000110 => 1
[5,3]
=> 1001000 => 1
[5,2,1]
=> 10001010 => 1
Description
The number of leading ones in a binary word.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,2,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,1,2] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,2] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,1,3] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [5,1] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,4,1] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,1,3] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,1,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [1,1,4] => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [6,1] => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,7] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,5,1] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,4,1] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [1,3,2] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,3] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,1,4] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [2,1,3] => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [1,1,5] => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [7,1] => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,8] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,6,1] => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,5,1] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [1,4,2] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,4,1] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,3,1,1] => 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,7,2] => [2,1,7] => ? = 2
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[8,6,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
Description
The first part of an integer composition.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00109: Permutations descent wordBinary words
St000326: Binary words ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 11 => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 110 => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 101 => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 10000 => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1100 => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 110 => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 101 => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 011 => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1001 => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00001 => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 100000 => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 11000 => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1100 => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1010 => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0100 => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 111 => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1001 => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0101 => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 10001 => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 000001 => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 1000000 => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 110000 => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 11000 => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 10100 => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1100 => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1110 => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1001 => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0110 => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 1010 => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1101 => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 10001 => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0011 => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 01001 => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => 100001 => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 0000001 => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 10000000 => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => 1100000 => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 110000 => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 101000 => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 11000 => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 11100 => 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ? => ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ? => ? = 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? => ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? => ? = 2
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [10,5,1,2,3,4,6,7,8,9] => ? => ? = 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [9,5,2,1,3,4,6,7,8] => ? => ? = 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [9,5,3,1,2,4,6,7,8] => ? => ? = 1
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[7,5,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,2,1,4,5,7] => ? => ? = 1
[8,6,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [9,6,4,1,2,3,5,7,8] => ? => ? = 1
[7,4,3,2]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,1,2,6,7] => ? => ? = 1
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[7,7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,0,0]
=> [8,9,4,1,2,3,5,6,7] => ? => ? = 2
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.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% values known / values provided: 88%distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[7,5,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[8,6,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ?
=> ? = 1
[8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ?
=> ? = 1
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ?
=> ?
=> ? = 1
[7,7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ?
=> ? = 2
[3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[5,5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[5,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[5,5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
Description
The number of up steps after the last double rise of a Dyck path.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00314: Integer compositions Foata bijectionInteger compositions
St000383: Integer compositions ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [1,2] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,3] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,2,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,4] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [5,1] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,1,2] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [1,5] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [6,1] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,4,1] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [2,3] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [4,1,1] => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [1,6] => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [7,1] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,5,1] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,4,1] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [3,2,1] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,2,1] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,2,1,1] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [4,1,1] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [3,1,2] => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [5,1,1] => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [1,7] => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [8,1] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,6,1] => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,5,1] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [4,2,1] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,4,1] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,3,1] => 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => [5,1,2] => ? = 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => ? => ? = 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,4,3] => [4,1,3] => ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,6,2] => [6,1,2] => ? = 2
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => ? => ? = 10
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4] => [1,3,4] => ? = 4
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,5,3] => [5,1,3] => ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,7,2] => [7,1,2] => ? = 2
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => [6,2] => ? = 2
[2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => [2,6] => ? = 6
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,5] => [1,2,5] => ? = 5
[3,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,4,3] => [4,1,3] => ? = 3
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,5,1] => [2,5,1] => ? = 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => [1,1,6] => ? = 6
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,2] => [7,2] => ? = 2
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,4,2] => [4,2,2] => ? = 2
[2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,7] => [2,7] => ? = 7
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ? => ? = 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ? => ? = 1
[7,5,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => [5,3] => ? = 3
[3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => [3,5] => ? = 5
[8,6,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[7,4,3,2]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[6,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [4,1,1,2] => [1,1,4,2] => ? = 2
[6,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,5,1] => [2,5,1] => ? = 1
[3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => [2,1,5] => ? = 5
[3,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,4] => [2,2,4] => ? = 4
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? => ? => ? = 1
[7,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[7,5,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[7,5,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[4,4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => [3,1,4] => ? = 4
[3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => [2,1,5] => ? = 5
[4,4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => [3,1,4] => ? = 4
[7,6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,6] => [3,6] => ? = 6
[4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5] => [4,5] => ? = 5
[5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,4] => [5,4] => ? = 4
[6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [6,3] => [6,3] => ? = 3
[5,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,4,1] => [4,4,1] => ? = 1
[5,5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,1,4] => [4,1,4] => ? = 4
Description
The last part of an integer composition.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000617: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ?
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ?
=> ? = 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ?
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ?
=> ? = 2
[4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [10,5,1,2,3,4,6,7,8,9] => ?
=> ? = 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [9,5,2,1,3,4,6,7,8] => ?
=> ? = 1
[4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,1,2,3] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[3,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,5,6,2,3,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [9,5,3,1,2,4,6,7,8] => ?
=> ? = 1
[4,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,8,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[3,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,8,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 3
[8,6,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [9,6,4,1,2,3,5,7,8] => ?
=> ? = 1
[7,4,3,2]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,1,2,6,7] => ?
=> ? = 1
[4,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,8,1,2,3] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,2,1,3] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[3,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,3,8,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4
[8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ? => ?
=> ? = 1
[7,7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,0,0]
=> [8,9,4,1,2,3,5,6,7] => ?
=> ? = 2
[4,4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,1,2,3] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 2
[3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,3,1,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,7,8,9,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 6
[4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [5,6,7,8,9,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,7,8,9,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 4
[6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [7,8,9,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 3
[5,5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 5
Description
The number of global maxima of a Dyck path.
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001038: Dyck paths ⟶ ℤResult quality: 70% values known / values provided: 84%distinct values known / distinct values provided: 70%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1
[2]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1]
=> [2]
=> []
=> []
=> ? = 2
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 3
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 4
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 5
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 6
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 7
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2,2]
=> [4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2,2,1,1]
=> [5,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 8
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 9
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 10
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,1]
=> [3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[8,4,3]
=> [3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,4,3,1]
=> [4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[8,6,2]
=> [3,3,2,2,2,2,1,1]
=> [3,2,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[7,4,3,2]
=> [4,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[6,2,2,2,2,2]
=> [6,6,1,1,1,1]
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [3,3,2,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[7,7,3]
=> [3,3,3,2,2,2,2]
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
[7,6,3,1]
=> [4,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[7,5,4,1]
=> [4,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [4,3,2,2,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[5,3,3,3,3]
=> [5,5,5,1,1]
=> [5,5,1,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[6,5,4,3,1]
=> [5,4,4,3,2,1]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 1
[6,6,5,4,3,2]
=> [6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[6,5,5,4,3,2]
=> [6,6,5,4,3,1]
=> [6,5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[6,5,4,4,3,2]
=> [6,6,5,4,2,1]
=> ?
=> ?
=> ? = 1
[6,5,4,3,3,2]
=> ?
=> ?
=> ?
=> ? = 1
[6,5,4,3,2,2]
=> ?
=> ?
=> ?
=> ? = 1
[6,6,5,4,3,1]
=> [6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[6,5,5,4,3,1]
=> [6,5,5,4,3,1]
=> [5,5,4,3,1]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000993: Integer partitions ⟶ ℤResult quality: 70% values known / values provided: 84%distinct values known / distinct values provided: 70%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1
[2]
=> [1,1]
=> [1]
=> [1]
=> ? = 1
[1,1]
=> [2]
=> []
=> []
=> ? = 2
[3]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[2,1]
=> [2,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 3
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 4
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 5
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 6
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 7
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [2,2,1]
=> 2
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2,2]
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 4
[2,2,2,1,1]
=> [5,3]
=> [3]
=> [1,1,1]
=> 3
[2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 8
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 9
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 10
[8,6,2]
=> [3,3,2,2,2,2,1,1]
=> [3,2,2,2,2,1,1]
=> [7,5,1]
=> ? = 1
[8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [7,4,2]
=> ? = 1
[6,6,4]
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 2
[6,5,5]
=> [3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> ? = 1
[8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [3,3,2,2,2,1,1]
=> [7,5,2]
=> ? = 1
[7,7,3]
=> [3,3,3,2,2,2,2]
=> [3,3,2,2,2,2]
=> [6,6,2]
=> ? = 2
[7,6,3,1]
=> [4,3,3,2,2,2,1]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[7,5,4,1]
=> [4,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,4,3]
=> ? = 1
[7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [4,3,2,2,1,1]
=> [6,4,2,1]
=> ? = 1
[6,6,5]
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 2
[5,5,5,2]
=> [4,4,3,3,3]
=> [4,3,3,3]
=> [4,4,4,1]
=> ? = 3
[5,5,4,3]
=> [4,4,4,3,2]
=> [4,4,3,2]
=> [4,4,3,2]
=> ? = 2
[5,4,4,4]
=> [4,4,4,4,1]
=> [4,4,4,1]
=> [4,3,3,3]
=> ? = 1
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 1
[5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,4,3,2]
=> [4,4,3,2,1]
=> ? = 2
[5,4,4,3,2]
=> [5,5,4,3,1]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> ? = 1
[5,5,3,3,2]
=> [5,5,4,2,2]
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 2
[5,5,4,2,2]
=> [5,5,3,3,2]
=> [5,3,3,2]
=> [4,4,3,1,1]
=> ? = 2
[6,5,4,3,1]
=> [5,4,4,3,2,1]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> ? = 1
[5,5,4,3,1]
=> [5,4,4,3,2]
=> [4,4,3,2]
=> [4,4,3,2]
=> ? = 2
[6,5,4,2,1]
=> [5,4,3,3,2,1]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> ? = 1
[6,6,5,4,3,2]
=> [6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 2
[6,5,5,4,3,2]
=> [6,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 1
[6,5,4,4,3,2]
=> [6,6,5,4,2,1]
=> ?
=> ?
=> ? = 1
[6,5,4,3,3,2]
=> ?
=> ?
=> ?
=> ? = 1
[6,5,4,3,2,2]
=> ?
=> ?
=> ?
=> ? = 1
[6,6,5,4,3,1]
=> [6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 2
[6,5,5,4,3,1]
=> [6,5,5,4,3,1]
=> [5,5,4,3,1]
=> [5,4,4,3,2]
=> ? = 1
[6,6,5,3,2,1]
=> ?
=> ?
=> ?
=> ? = 2
[6,6,4,3,2,1]
=> [6,5,4,3,2,2]
=> [5,4,3,2,2]
=> [5,5,3,2,1]
=> ? = 2
[5,4,4,4,3]
=> ?
=> ?
=> ?
=> ? = 1
Description
The multiplicity of the largest part of an integer partition.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St001107: Dyck paths ⟶ ℤResult quality: 60% values known / values provided: 77%distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4 = 5 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 1 - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> ? = 3 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 4 - 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 5 - 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2 - 1
[3,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 1 - 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 - 1
[2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? = 7 - 1
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 1 - 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[6,6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5 - 1
[3,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 1
[8,6,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ?
=> ? = 1 - 1
[8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ?
=> ? = 1 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
Mp00091: Set partitions rotate increasingSet partitions
St000823: Set partitions ⟶ ℤResult quality: 70% values known / values provided: 73%distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 1
[2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,4},{2,3}}
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> {{1},{2,3,5},{4}}
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2,3},{4}}
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2},{3,4}}
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> {{1,3,4,5,6},{2}}
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> {{1},{2,3,4,5,6,7}}
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> {{1},{2,3,4,6},{5}}
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,4,5},{3}}
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,5},{2,3,4}}
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,3},{2},{4,5}}
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,4,5},{2,3}}
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,3,5},{2},{4}}
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> {{1,3,4,5},{2},{6}}
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7}}
=> {{1,3,4,5,6,7},{2}}
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7,8}}
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,6},{5},{7}}
=> {{1},{2,3,4,5,7},{6}}
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> {{1},{2,3,6},{4,5}}
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> {{1},{2,3,5,6},{4}}
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2,3,4},{5}}
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2},{3,4,5}}
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,4},{3}}
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> {{1,3,4},{2},{5,6}}
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> {{1,3,4,6},{2},{5}}
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,7},{6}}
=> {{1,3,4,5,6},{2},{7}}
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> {{1,3,4,5,6,7,8},{2}}
=> 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> {{1},{2,3,4,5,6,8},{7}}
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,6},{4,5},{7}}
=> {{1},{2,3,4,7},{5,6}}
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,6},{4},{7}}
=> {{1},{2,3,4,6,7},{5}}
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> {{1,5},{2,3,4},{6}}
=> {{1},{2,6},{3,4,5}}
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> {{1,2,5},{3},{4},{6}}
=> {{1},{2,3,6},{4},{5}}
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> {{1,3,4,5},{2},{6}}
=> {{1},{2,4,5,6},{3}}
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 8
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> {{1},{2,3,4,5,6,7,9},{8}}
=> ? = 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5,6},{8}}
=> {{1},{2,3,4,5,8},{6,7}}
=> ? = 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> {{1,3,4,5,6},{2},{7,8}}
=> ? = 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,3,4,5,6,8},{2},{7}}
=> ? = 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8},{2},{9}}
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> {{1,3,4,5,6,7,8,9,10},{2}}
=> ? = 9
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> {{1},{2,3,4,5,6,7,8,9,10,11}}
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> {{1},{2,3,4,5,6,7,8,10},{9}}
=> ? = 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,8},{6,7},{9}}
=> {{1},{2,3,4,5,6,9},{7,8}}
=> ? = 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7,8},{6},{9}}
=> {{1},{2,3,4,5,6,8,9},{7}}
=> ? = 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4,5,6},{8}}
=> {{1},{2,3,4,8},{5,6,7}}
=> ? = 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5},{6},{8}}
=> {{1},{2,3,4,5,8},{6},{7}}
=> ? = 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6,7}}
=> {{1,3,4,5},{2},{6,7,8}}
=> ? = 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6},{7}}
=> {{1,3,4,5,6},{2},{7},{8}}
=> ? = 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7,8}}
=> {{1,3,4,5,6,7},{2},{8,9}}
=> ? = 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,3,4,5,7,8},{2},{6}}
=> ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8,9},{7}}
=> {{1,3,4,5,6,7,9},{2},{8}}
=> ? = 2
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,10},{9}}
=> {{1,3,4,5,6,7,8,9},{2},{10}}
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10,11}}
=> {{1,3,4,5,6,7,8,9,10,11},{2}}
=> ? = 10
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,9},{7,8},{10}}
=> ?
=> ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,8},{6},{7},{9}}
=> ?
=> ? = 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4,5,6},{8}}
=> {{1},{2,3,8},{4,5,6,7}}
=> ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,7},{4},{6},{8}}
=> {{1},{2,3,4,6,8},{5},{7}}
=> ? = 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,7},{6}}
=> {{1,3,4,5},{2},{6,8},{7}}
=> ? = 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7},{8}}
=> {{1,3,4,5,6,7},{2},{8},{9}}
=> ? = 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,5,6,7,8},{4}}
=> {{1,3,4,6,7,8},{2},{5}}
=> ? = 4
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8,9},{6}}
=> ?
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9,10},{8}}
=> ?
=> ? = 2
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,4,5,6},{8}}
=> {{1},{2,8},{3,4,5,6,7}}
=> ? = 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,9},{5,6,7,8},{10}}
=> ?
=> ? = 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,8},{4,5,7},{6},{9}}
=> ?
=> ? = 1
[6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,6},{5},{7,8}}
=> {{1,8},{2,3,4,5,7},{6}}
=> ? = 2
[3,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> {{1},{2,3,6,7,8},{4,5}}
=> ?
=> ? = 3
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2},{3,4,5,6,8},{7}}
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,8},{4,7},{5,6},{9}}
=> ?
=> ? = 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7},{8,9}}
=> {{1,9},{2,3,4,5,6,7,8}}
=> ? = 2
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,6},{4},{5},{8}}
=> ?
=> ? = 1
[6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,6},{4,5},{7,8}}
=> {{1,8},{2,3,4,7},{5,6}}
=> ? = 2
[3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,2},{3,4,5,7,8},{6}}
=> {{1,4,5,6,8},{2,3},{7}}
=> ? = 2
[2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9}}
=> {{1,4,5,6,7,8,9},{2,3}}
=> ? = 7
[8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[8,4,3]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 1
[7,5,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,6},{4},{5},{8}}
=> ?
=> ? = 1
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> {{1,2,7},{3,5},{4},{6},{8}}
=> ?
=> ? = 1
[6,6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> {{1,2,6},{3,4,5},{7,8}}
=> {{1,8},{2,3,7},{4,5,6}}
=> ? = 2
Description
The number of unsplittable factors of the set partition. Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks $$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$ if $k\leq \ell$ and $$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$ otherwise.
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000918The 2-limited packing 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. St000439The position of the first down step of a Dyck path. St000234The number of global ascents of a permutation. St000759The smallest missing part in an integer partition. St000733The row containing the largest entry of a standard tableau. St000989The number of final rises of a permutation. St000546The number of global descents of a permutation. St000260The radius of a connected graph. St000990The first ascent of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000007The number of saliances of the permutation. St000883The number of longest increasing subsequences of a permutation. St000971The smallest closer of a set partition. St000654The first descent 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. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000025The number of initial rises of a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000911The number of maximal antichains of maximal size in a poset. St000765The number of weak records in an integer composition. St000054The first entry of the permutation. St000657The smallest part of an integer composition. St000542The number of left-to-right-minima 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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000026The position of the first return of a Dyck path. St001481The minimal height of a peak of a Dyck path. St000090The variation of a composition. 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. St000069The number of maximal elements of a poset. St001050The number of terminal closers of a set partition. St001075The minimal size of a block of a set partition. St000909The number of maximal chains of maximal size in a poset. St000363The number of minimal vertex covers of a graph. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000991The number of right-to-left minima of a permutation. St000908The length of the shortest maximal antichain in a poset. St000264The girth of a graph, which is not a tree. St000740The last entry of a permutation. St000314The number of left-to-right-maxima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.