Your data matches 76 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000745
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
 => [[1]]
 => 1
[1,2] => [1,2] => [[1,2]]
 => [[1,2]]
 => 1
[2,1] => [2,1] => [[1],[2]]
 => [[1],[2]]
 => 2
[1,2,3] => [1,2,3] => [[1,2,3]]
 => [[1,2,3]]
 => 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => [[1],[2],[3]]
 => 3
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => [[1,2,3,4]]
 => 1
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 1
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 2
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
 => [[1,3],[2],[4]]
 => 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 1
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 2
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [[1,2,5],[3],[4]]
 => 1
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000297
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => => ? = 1 - 1
[1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => [2,1] => 1 => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[2,1,3] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[2,3,1] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[3,1,2] => [3,1,2] => 01 => 10 => 1 = 2 - 1
[3,2,1] => [3,2,1] => 11 => 11 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[1,3,4,2] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => 100 => 001 => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[2,3,1,4] => [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[2,3,4,1] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => 001 => 100 => 1 = 2 - 1
[2,4,3,1] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[3,1,2,4] => [3,1,2,4] => 010 => 010 => 0 = 1 - 1
[3,1,4,2] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 110 => 011 => 0 = 1 - 1
[3,2,4,1] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[3,4,1,2] => [2,4,1,3] => 001 => 100 => 1 = 2 - 1
[3,4,2,1] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[4,1,2,3] => [4,1,2,3] => 001 => 100 => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => 011 => 110 => 2 = 3 - 1
[4,2,1,3] => [4,2,1,3] => 101 => 101 => 1 = 2 - 1
[4,2,3,1] => [4,1,3,2] => 011 => 110 => 2 = 3 - 1
[4,3,1,2] => [4,3,1,2] => 011 => 110 => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => 111 => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010 => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,3,4,2,5] => [1,2,4,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,4,2] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,4,2,5,3] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 0110 => 0 = 1 - 1
[1,4,3,5,2] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,4,5,2,3] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,4,5,3,2] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[6,2,8,4,5,1,7,3] => [5,1,8,2,4,3,7,6] => ? => ? => ? = 3 - 1
[6,5,7,8,2,1,4,3] => [2,1,5,8,4,3,7,6] => ? => ? => ? = 3 - 1
[6,4,7,8,2,1,5,3] => [2,1,5,8,4,3,7,6] => ? => ? => ? = 3 - 1
[3,2,5,4,8,1,6,7] => [2,1,5,4,8,3,6,7] => ? => ? => ? = 2 - 1
[5,1,8,2,7,4,6,3] => [4,1,8,2,7,3,6,5] => ? => ? => ? = 4 - 1
[7,8,3,1,6,5,2,4] => [4,8,3,1,7,6,2,5] => ? => ? => ? = 4 - 1
[5,6,3,2,4,1,8,7] => [3,6,2,1,5,4,8,7] => ? => ? => ? = 2 - 1
[2,1,7,8,5,4,6,3] => [2,1,5,8,4,3,7,6] => ? => ? => ? = 3 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000340
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 5 - 1
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 1
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 3 - 1
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 6 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[6,5,7,8,3,4,1,2] => [2,1,4,3,8,7,5,6] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[8,4,3,5,6,7,1,2] => [2,1,7,6,5,3,4,8] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3 - 1
[6,3,4,5,7,8,1,2] => [2,1,8,7,5,4,3,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2 - 1
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2 - 1
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
 => ? = 6 - 1
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4 - 1
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 3 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4 - 1
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => ? = 3 - 1
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,3,2,1,4] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[7,8,6,5,2,1,3,4] => [4,3,1,2,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[7,8,5,6,2,1,3,4] => [4,3,1,2,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => ? = 3 - 1
[5,6,7,8,2,1,3,4] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 4 - 1
[6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 3 - 1
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 4 - 1
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 3 - 1
[7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 - 1
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 - 1
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 - 1
[7,8,6,1,2,3,4,5] => [5,4,3,2,1,6,8,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? = 3 - 1
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 3 - 1
[7,6,8,1,2,3,4,5] => [5,4,3,2,1,8,6,7] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 - 1
[8,7,5,2,1,3,4,6] => [6,4,3,1,2,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[7,8,5,2,1,3,4,6] => [6,4,3,1,2,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 - 1
[8,7,5,1,2,3,4,6] => [6,4,3,2,1,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[7,8,5,1,2,3,4,6] => [6,4,3,2,1,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 - 1
[7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 - 1
[8,6,5,4,3,2,1,7] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[8,4,5,6,3,2,1,7] => [7,1,2,3,6,5,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[8,6,5,4,3,1,2,7] => [7,2,1,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[8,6,5,4,2,1,3,7] => [7,3,1,2,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[8,6,5,2,3,1,4,7] => [7,4,1,3,2,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[8,5,6,2,3,1,4,7] => [7,4,1,3,2,6,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
 => 1
[1,2] => [2,1] => [2,1] => [2]
 => 1
[2,1] => [1,2] => [1,2] => [1,1]
 => 2
[1,2,3] => [3,2,1] => [2,3,1] => [3]
 => 1
[1,3,2] => [2,3,1] => [3,2,1] => [2,1]
 => 2
[2,1,3] => [3,1,2] => [3,1,2] => [3]
 => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1]
 => 2
[3,1,2] => [2,1,3] => [2,1,3] => [2,1]
 => 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
 => 3
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [4]
 => 1
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [3,1]
 => 2
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4]
 => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [3,1]
 => 2
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [2,2]
 => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 3
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [4]
 => 1
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [3,1]
 => 2
[2,3,1,4] => [4,1,3,2] => [4,3,1,2] => [4]
 => 1
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [3,1]
 => 2
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [2,2]
 => 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 3
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [4]
 => 1
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,1]
 => 2
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4]
 => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,1]
 => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 3
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [3,1]
 => 2
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 3
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1]
 => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 3
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 3
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => [5]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => [4,1]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => [5]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => [4,1]
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => [3,2]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => [3,1,1]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => [5]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => [4,1]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,1,2] => [5]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => [4,1]
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,1,2] => [3,2]
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => [3,1,1]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => [5]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => [4,1]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => [5]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => [4,1]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => [3,2]
 => 2
[7,8,6,5,2,1,3,4] => [4,3,1,2,5,6,8,7] => [3,1,4,2,5,6,8,7] => ?
 => ? = 4
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => [3,1,5,2,4,6,7,8] => ?
 => ? = 4
[8,7,5,2,1,3,4,6] => [6,4,3,1,2,5,7,8] => [3,1,4,6,2,5,7,8] => ?
 => ? = 3
[7,8,5,1,2,3,4,6] => [6,4,3,2,1,5,8,7] => [2,3,4,6,1,5,8,7] => ?
 => ? = 2
[8,4,5,6,3,2,1,7] => [7,1,2,3,6,5,4,8] => [7,1,2,5,6,3,4,8] => ?
 => ? = 2
[8,6,5,2,3,1,4,7] => [7,4,1,3,2,5,6,8] => [4,3,1,7,2,5,6,8] => ?
 => ? = 2
[8,6,4,3,2,1,5,7] => [7,5,1,2,3,4,6,8] => [5,1,2,3,7,4,6,8] => ?
 => ? = 2
[2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [1,6,3,4,5,8,7,2] => ?
 => ? = 6
[2,3,5,6,8,7,4,1] => [1,4,7,8,6,5,3,2] => [1,3,5,4,6,8,7,2] => ?
 => ? = 4
[2,3,5,6,7,8,4,1] => [1,4,8,7,6,5,3,2] => [1,3,5,4,6,7,8,2] => ?
 => ? = 3
[3,4,2,5,7,8,6,1] => [1,6,8,7,5,2,4,3] => [1,5,4,2,7,6,8,3] => ?
 => ? = 3
[3,2,6,5,7,4,8,1] => [1,8,4,7,5,6,2,3] => [1,6,2,8,7,5,4,3] => ?
 => ? = 2
[1,4,3,8,7,6,5,2] => [2,5,6,7,8,3,4,1] => [4,2,8,3,5,6,7,1] => ?
 => ? = 5
[1,6,5,4,3,8,7,2] => [2,7,8,3,4,5,6,1] => [6,2,8,3,4,5,7,1] => ?
 => ? = 3
[2,1,6,7,5,8,4,3] => [3,4,8,5,7,6,1,2] => [6,1,3,4,7,8,2,5] => ?
 => ? = 3
[2,1,5,4,7,6,8,3] => [3,8,6,7,4,5,1,2] => [5,1,3,7,4,8,6,2] => ?
 => ? = 2
[2,4,3,1,6,8,7,5] => [5,7,8,6,1,3,4,2] => [6,4,1,3,5,8,7,2] => ?
 => ? = 3
[2,1,4,3,7,6,8,5] => [5,8,6,7,3,4,1,2] => [4,1,7,3,5,8,6,2] => ?
 => ? = 2
[4,5,3,6,2,1,8,7] => [7,8,1,2,6,3,5,4] => [8,1,6,5,3,2,7,4] => ?
 => ? = 2
[3,2,5,4,6,1,8,7] => [7,8,1,6,4,5,2,3] => [8,5,2,6,4,1,7,3] => ?
 => ? = 2
[2,1,5,4,6,3,8,7] => [7,8,3,6,4,5,1,2] => [5,1,8,6,4,3,7,2] => ?
 => ? = 2
[2,3,4,1,6,5,8,7] => [7,8,5,6,1,4,3,2] => [6,3,4,1,8,5,7,2] => ?
 => ? = 2
[1,8,5,4,7,6,3,2] => [2,3,6,7,4,5,8,1] => [8,2,3,7,4,6,5,1] => ?
 => ? = 5
[2,3,4,8,5,6,7,1] => [1,7,6,5,8,4,3,2] => [1,3,4,8,6,7,5,2] => ?
 => ? = 3
[2,4,1,6,7,3,8,5] => [5,8,3,7,6,1,4,2] => [6,4,7,1,5,8,2,3] => ?
 => ? = 2
[2,1,4,3,7,5,8,6] => [6,8,5,7,3,4,1,2] => [4,1,7,3,8,6,5,2] => ?
 => ? = 2
[2,1,4,3,8,5,6,7] => [7,6,5,8,3,4,1,2] => [4,1,8,3,6,7,5,2] => ?
 => ? = 2
[2,1,6,3,7,4,8,5] => [5,8,4,7,3,6,1,2] => [6,1,7,8,5,3,4,2] => ?
 => ? = 2
[3,1,4,6,8,2,5,7] => [7,5,2,8,6,4,1,3] => [4,6,1,5,8,7,2,3] => ?
 => ? = 2
[3,1,5,2,6,4,8,7] => [7,8,4,6,2,5,1,3] => [5,6,1,8,2,4,7,3] => ?
 => ? = 2
[3,1,5,2,7,4,8,6] => [6,8,4,7,2,5,1,3] => [5,7,1,8,2,6,4,3] => ?
 => ? = 2
[3,1,5,2,8,4,6,7] => [7,6,4,8,2,5,1,3] => [5,8,1,6,2,7,4,3] => ?
 => ? = 2
[4,6,1,7,2,3,5,8] => [8,5,3,2,7,1,6,4] => [7,3,5,8,6,1,4,2] => ?
 => ? = 1
[1,4,2,6,7,8,3,5] => [5,3,8,7,6,2,4,1] => [4,6,5,2,3,7,8,1] => ?
 => ? = 2
[4,1,2,6,3,8,5,7] => [7,5,8,3,6,2,1,4] => [2,6,8,1,7,3,5,4] => ?
 => ? = 2
[5,1,6,2,3,8,4,7] => [7,4,8,3,2,6,1,5] => [6,3,8,7,1,2,4,5] => ?
 => ? = 2
[1,5,6,2,3,8,4,7] => [7,4,8,3,2,6,5,1] => [5,3,6,8,7,1,2,4] => ?
 => ? = 2
[5,7,8,1,2,3,4,6] => [6,4,3,2,1,8,7,5] => [2,3,4,6,7,1,8,5] => ?
 => ? = 2
[6,1,7,8,2,3,4,5] => [5,4,3,2,8,7,1,6] => [7,3,4,5,2,1,8,6] => ?
 => ? = 2
[6,5,8,1,2,3,4,7] => [7,4,3,2,1,8,5,6] => [2,3,4,7,8,5,1,6] => ?
 => ? = 2
[7,8,2,4,6,1,3,5] => [5,3,1,6,4,2,8,7] => [3,5,4,6,2,1,8,7] => ?
 => ? = 3
[4,3,2,6,8,1,5,7] => [7,5,1,8,6,2,3,4] => [5,7,2,3,6,8,4,1] => ?
 => ? = 2
[6,5,2,4,8,1,3,7] => [7,3,1,8,4,2,5,6] => [3,7,4,8,2,5,6,1] => ?
 => ? = 2
[8,7,2,4,6,1,3,5] => [5,3,1,6,4,2,7,8] => [3,5,4,6,2,1,7,8] => ?
 => ? = 4
[7,5,8,2,4,1,3,6] => [6,3,1,4,2,8,5,7] => [3,6,4,2,8,1,5,7] => ?
 => ? = 2
[6,5,3,8,2,7,1,4] => [4,1,7,2,8,3,5,6] => [4,7,8,1,3,5,2,6] => ?
 => ? = 3
[7,5,3,8,2,6,1,4] => [4,1,6,2,8,3,5,7] => [4,6,8,1,3,2,5,7] => ?
 => ? = 3
[8,5,4,7,2,6,1,3] => [3,1,6,2,7,4,5,8] => [3,6,1,7,4,2,5,8] => ?
 => ? = 4
[8,5,4,7,3,6,1,2] => [2,1,6,3,7,4,5,8] => [2,1,6,7,4,3,5,8] => ?
 => ? = 4
[4,1,2,8,7,3,5,6] => [6,5,3,7,8,2,1,4] => [2,8,5,1,6,3,7,4] => ?
 => ? = 3
Description
The length of the partition.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
 => [1,0]
 => 1
[1,2] => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[2,1] => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 2
[1,2,3] => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 1
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 2
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,1,3,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,3,1,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[3,1,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 1
[3,1,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[3,2,1,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 1
[3,2,4,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 3
[4,2,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 3
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 3
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,3,4,2,1] => [4,3,2,1,6,5,7,8] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[5,6,7,8,2,3,4,1] => [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[5,6,7,8,4,1,2,3] => [4,3,2,1,5,8,7,6] => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[5,6,7,8,3,2,1,4] => [4,3,2,1,6,7,8,5] => [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2
[7,8,6,5,2,1,3,4] => [2,1,3,4,7,8,6,5] => [[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
 => [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 4
[7,8,6,5,1,2,3,4] => [2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
 => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[8,6,7,5,1,2,3,4] => [1,3,2,4,8,7,6,5] => [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[6,7,8,5,1,2,3,4] => [3,2,1,4,8,7,6,5] => [[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,7,5,6,1,2,3,4] => [1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[7,5,6,8,1,2,3,4] => [2,4,3,1,8,7,6,5] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[8,7,6,4,2,1,3,5] => [1,2,3,5,7,8,6,4] => [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 4
[7,8,5,4,2,1,3,6] => [2,1,4,5,7,8,6,3] => [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
 => [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2
[8,7,5,2,1,3,4,6] => [1,2,4,7,8,6,5,3] => [.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 3
[7,8,5,2,1,3,4,6] => [2,1,4,7,8,6,5,3] => [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
 => [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[8,7,5,1,2,3,4,6] => [1,2,4,8,7,6,5,3] => [.,[.,[[.,[[[[.,.],.],.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 3
[8,6,5,4,2,1,3,7] => [1,3,4,5,7,8,6,2] => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2
[8,6,5,2,3,1,4,7] => [1,3,4,7,6,8,5,2] => [.,[[.,[.,[[[.,.],[.,.]],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 2
[8,5,6,2,3,1,4,7] => [1,4,3,7,6,8,5,2] => [.,[[[.,.],[[[.,.],[.,.]],.]],.]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 2
[8,6,5,1,2,3,4,7] => [1,3,4,8,7,6,5,2] => [.,[[.,[.,[[[[.,.],.],.],.]]],.]]
 => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[8,6,4,3,2,1,5,7] => [1,3,5,6,7,8,4,2] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2
[8,6,3,4,2,1,5,7] => [1,3,6,5,7,8,4,2] => [.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
 => [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 2
[6,5,7,3,2,4,1,8] => [3,4,2,6,7,5,8,1] => [[[.,[.,.]],[[.,[.,.]],[.,.]]],.]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
 => ? = 1
[6,5,7,1,2,3,4,8] => [3,4,2,8,7,6,5,1] => [[[.,[.,.]],[[[[.,.],.],.],.]],.]
 => [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,4,2,5,7,8,6,1] => [6,5,7,4,2,1,3,8] => [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 3
[1,4,3,6,5,8,7,2] => [8,5,6,3,4,1,2,7] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 3
[2,1,6,7,8,5,4,3] => [7,8,3,2,1,4,5,6] => [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[2,1,6,7,5,8,4,3] => [7,8,3,2,4,1,5,6] => [[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 3
[2,1,5,6,7,8,4,3] => [7,8,4,3,2,1,5,6] => [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 3
[4,3,2,1,7,6,8,5] => [5,6,7,8,2,3,1,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
[4,3,2,1,6,7,8,5] => [5,6,7,8,3,2,1,4] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
[3,4,2,1,7,8,6,5] => [6,5,7,8,2,1,3,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 3
[2,4,3,1,6,8,7,5] => [7,5,6,8,3,1,2,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 3
[3,2,4,1,8,7,6,5] => [6,7,5,8,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
[3,2,4,1,6,8,7,5] => [6,7,5,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 3
[2,1,4,3,8,7,6,5] => [7,8,5,6,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
[2,1,4,3,7,8,6,5] => [7,8,5,6,2,1,3,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 3
[2,1,4,3,6,8,7,5] => [7,8,5,6,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 3
[2,1,4,3,6,7,8,5] => [7,8,5,6,3,2,1,4] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 2
[2,1,3,4,8,7,6,5] => [7,8,6,5,1,2,3,4] => [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[4,3,2,1,5,7,8,6] => [5,6,7,8,4,2,1,3] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
[3,2,1,4,5,8,7,6] => [6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 3
[4,5,3,6,2,1,8,7] => [5,4,6,3,7,8,1,2] => [[[[.,.],[.,.]],[.,[.,.]]],[.,.]]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
[2,1,4,5,6,3,8,7] => [7,8,5,4,3,6,1,2] => [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 2
[4,3,2,1,6,5,8,7] => [5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 2
[2,3,4,1,6,5,8,7] => [7,6,5,8,3,4,1,2] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 2
[4,3,2,1,5,6,8,7] => [5,6,7,8,4,3,1,2] => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 2
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000925
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1,0]
 => {{1}}
 => ? = 1
[1,2] => [2,1] => [1,1,0,0]
 => {{1,2}}
 => 1
[2,1] => [1,2] => [1,0,1,0]
 => {{1},{2}}
 => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 3
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 3
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 8
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3},{4},{5,6,7,8}}
 => ? = 5
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1},{2},{3,4},{5,6},{7},{8}}
 => ? = 6
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4},{5,6,7,8}}
 => ? = 4
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4},{5},{6,7},{8}}
 => ? = 6
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4},{5,6,7,8}}
 => ? = 3
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 6
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 4
[6,5,7,8,3,4,1,2] => [2,1,4,3,8,7,5,6] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 3
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 3
[8,4,3,5,6,7,1,2] => [2,1,7,6,5,3,4,8] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 3
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6},{7},{8}}
 => ? = 6
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 4
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 4
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 3
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2
[5,6,7,8,3,2,1,4] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2
[7,8,6,5,2,1,3,4] => [4,3,1,2,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 4
[7,8,5,6,2,1,3,4] => [4,3,1,2,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 3
[5,6,7,8,2,1,3,4] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 5
[7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 4
[8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4},{5},{6,7},{8}}
 => ? = 4
[6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 3
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 4
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3
[7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 4
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 4
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 4
[7,8,6,1,2,3,4,5] => [5,4,3,2,1,6,8,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 3
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 3
[7,6,8,1,2,3,4,5] => [5,4,3,2,1,8,6,7] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 2
[6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 2
[8,7,5,2,1,3,4,6] => [6,4,3,1,2,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 3
[8,7,5,1,2,3,4,6] => [6,4,3,2,1,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 3
[3,5,8,7,6,4,2,1] => [1,2,4,6,7,8,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => {{1},{2},{3,5,8},{4},{6},{7}}
 => ? = 6
[3,5,6,7,8,4,2,1] => [1,2,4,8,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1},{2},{3,5,6,7,8},{4}}
 => ? = 4
[2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => {{1},{2,6,8},{3},{4},{5},{7}}
 => ? = 6
[2,5,6,7,8,4,3,1] => [1,3,4,8,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => {{1},{2,5,6,7,8},{3},{4}}
 => ? = 4
[2,3,5,6,8,7,4,1] => [1,4,7,8,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => {{1},{2,3,5,6,8},{4},{7}}
 => ? = 4
[1,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 7
[1,7,8,6,5,4,3,2] => [2,3,4,5,6,8,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,7,8},{2},{3},{4},{5},{6}}
 => ? = 6
[1,5,6,7,8,4,3,2] => [2,3,4,8,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,5,6,7,8},{2},{3},{4}}
 => ? = 4
[1,3,5,7,8,6,4,2] => [2,4,6,8,7,5,3,1] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => {{1,3,5,7,8},{2},{4},{6}}
 => ? = 4
[1,4,3,8,7,6,5,2] => [2,5,6,7,8,3,4,1] => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => {{1,3,4,8},{2},{5},{6},{7}}
 => ? = 5
Description
The number of topologically connected components of a set partition. For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$. The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000159
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 87%●distinct values known / distinct values provided: 62%
Values
[1] => [1] => [1,0]
 => []
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
 => []
 => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
 => [1]
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 1 = 2 - 1
[4,6,5,3,2,1] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => ? = 5 - 1
[5,6,3,4,2,1] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => ? = 4 - 1
[5,6,4,2,3,1] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => ? = 4 - 1
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => ? = 4 - 1
[5,6,4,3,2,1] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => ? = 5 - 1
[6,3,5,4,2,1] => [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => ? = 5 - 1
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => ? = 4 - 1
[6,4,5,3,1,2] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => ? = 4 - 1
[6,4,5,3,2,1] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => ? = 5 - 1
[6,5,2,4,3,1] => [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => ? = 5 - 1
[6,5,3,4,1,2] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => ? = 4 - 1
[6,5,3,4,2,1] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => ? = 5 - 1
[6,5,4,1,3,2] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 5 - 1
[6,5,4,2,3,1] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => ? = 5 - 1
[6,5,4,3,1,2] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 5 - 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 6 - 1
[1,5,7,6,4,3,2] => [2,3,4,6,7,5,1] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,1]
 => ? = 5 - 1
[1,6,7,4,5,3,2] => [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,1]
 => ? = 4 - 1
[1,6,7,5,3,4,2] => [2,4,3,5,7,6,1] => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [4,4,3,1,1]
 => ? = 4 - 1
[1,6,7,5,4,2,3] => [3,2,4,5,7,6,1] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2]
 => ? = 4 - 1
[1,6,7,5,4,3,2] => [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,1]
 => ? = 5 - 1
[1,7,4,6,5,3,2] => [2,3,5,6,4,7,1] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,1]
 => ? = 5 - 1
[1,7,5,6,3,4,2] => [2,4,3,6,5,7,1] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,3,1,1]
 => ? = 4 - 1
[1,7,5,6,4,2,3] => [3,2,4,6,5,7,1] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2]
 => ? = 4 - 1
[1,7,5,6,4,3,2] => [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,1]
 => ? = 5 - 1
[1,7,6,3,5,4,2] => [2,4,5,3,6,7,1] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,1]
 => ? = 5 - 1
[1,7,6,4,5,2,3] => [3,2,5,4,6,7,1] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2]
 => ? = 4 - 1
[1,7,6,4,5,3,2] => [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,1]
 => ? = 5 - 1
[1,7,6,5,2,4,3] => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1]
 => ? = 5 - 1
[1,7,6,5,3,4,2] => [2,4,3,5,6,7,1] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,1]
 => ? = 5 - 1
[1,7,6,5,4,2,3] => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 5 - 1
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1]
 => ? = 6 - 1
[2,4,7,6,5,3,1] => [1,3,5,6,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,1,1]
 => ? = 5 - 1
[2,5,6,7,4,3,1] => [1,3,4,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1,1]
 => ? = 4 - 1
[2,5,7,6,3,4,1] => [1,4,3,6,7,5,2] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,3,1,1,1]
 => ? = 4 - 1
[2,5,7,6,4,3,1] => [1,3,4,6,7,5,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,1,1]
 => ? = 5 - 1
[2,6,5,7,4,3,1] => [1,3,4,7,5,6,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1,1]
 => ? = 4 - 1
[2,6,7,3,5,4,1] => [1,4,5,3,7,6,2] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,4,2,1,1,1]
 => ? = 4 - 1
[2,6,7,4,5,3,1] => [1,3,5,4,7,6,2] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,1,1]
 => ? = 4 - 1
[2,6,7,5,3,4,1] => [1,4,3,5,7,6,2] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,4,3,1,1,1]
 => ? = 4 - 1
[2,6,7,5,4,1,3] => [3,1,4,5,7,6,2] => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2]
 => ? = 4 - 1
[2,6,7,5,4,3,1] => [1,3,4,5,7,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,1,1]
 => ? = 5 - 1
[2,7,3,6,5,4,1] => [1,4,5,6,3,7,2] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,1,1]
 => ? = 5 - 1
[2,7,4,5,6,3,1] => [1,3,6,5,4,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,1,1]
 => ? = 4 - 1
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,1,1]
 => ? = 5 - 1
[2,7,5,4,6,3,1] => [1,3,6,4,5,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,1,1]
 => ? = 4 - 1
[2,7,5,6,3,4,1] => [1,4,3,6,5,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,3,1,1,1]
 => ? = 4 - 1
[2,7,5,6,4,1,3] => [3,1,4,6,5,7,2] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2]
 => ? = 4 - 1
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,1,1]
 => ? = 5 - 1
[2,7,6,3,4,5,1] => [1,5,4,3,6,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,1,1,1]
 => ? = 4 - 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
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1,0]
 => []
 => 1
[1,2] => [2,1] => [1,1,0,0]
 => []
 => 1
[2,1] => [1,2] => [1,0,1,0]
 => [1]
 => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 3
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2
[2,3,6,7,5,4,1] => [1,4,5,7,6,3,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1,1,1]
 => ? = 4
[2,3,7,5,6,4,1] => [1,4,6,5,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,2,1,1,1]
 => ? = 4
[2,3,7,6,4,5,1] => [1,5,4,6,7,3,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,1,1,1]
 => ? = 4
[2,3,7,6,5,4,1] => [1,4,5,6,7,3,2] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,1,1]
 => ? = 5
[2,4,5,7,6,3,1] => [1,3,6,7,5,4,2] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,2,2,1,1]
 => ? = 4
[2,4,7,5,6,3,1] => [1,3,6,5,7,4,2] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,1,1]
 => ? = 4
[2,4,7,6,3,5,1] => [1,5,3,6,7,4,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,1,1,1]
 => ? = 4
[2,4,7,6,5,3,1] => [1,3,5,6,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,1,1]
 => ? = 5
[2,5,4,7,6,3,1] => [1,3,6,7,4,5,2] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,2,2,1,1]
 => ? = 4
[2,5,6,7,3,4,1] => [1,4,3,7,6,5,2] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3,3,1,1,1]
 => ? = 3
[2,5,6,7,4,3,1] => [1,3,4,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1,1]
 => ? = 4
[2,5,7,3,6,4,1] => [1,4,6,3,7,5,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,2,1,1,1]
 => ? = 4
[2,5,7,4,6,3,1] => [1,3,6,4,7,5,2] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,1,1]
 => ? = 4
[2,5,7,6,3,4,1] => [1,4,3,6,7,5,2] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,3,1,1,1]
 => ? = 4
[2,5,7,6,4,3,1] => [1,3,4,6,7,5,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,1,1]
 => ? = 5
[2,6,3,7,5,4,1] => [1,4,5,7,3,6,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1,1,1]
 => ? = 4
[2,6,5,7,3,4,1] => [1,4,3,7,5,6,2] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3,3,1,1,1]
 => ? = 3
[2,6,5,7,4,3,1] => [1,3,4,7,5,6,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1,1]
 => ? = 4
[2,6,7,3,4,5,1] => [1,5,4,3,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,4,1,1,1,1]
 => ? = 3
[2,6,7,3,5,4,1] => [1,4,5,3,7,6,2] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,4,2,1,1,1]
 => ? = 4
[2,6,7,4,3,5,1] => [1,5,3,4,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,4,1,1,1,1]
 => ? = 3
[2,6,7,4,5,3,1] => [1,3,5,4,7,6,2] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,1,1]
 => ? = 4
[2,6,7,5,3,4,1] => [1,4,3,5,7,6,2] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,4,3,1,1,1]
 => ? = 4
[2,6,7,5,4,3,1] => [1,3,4,5,7,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,1,1]
 => ? = 5
[2,7,3,4,6,5,1] => [1,5,6,4,3,7,2] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [5,2,1,1,1,1]
 => ? = 4
[2,7,3,5,6,4,1] => [1,4,6,5,3,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,2,1,1,1]
 => ? = 4
[2,7,3,6,4,5,1] => [1,5,4,6,3,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,1,1,1]
 => ? = 4
[2,7,3,6,5,4,1] => [1,4,5,6,3,7,2] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,1,1]
 => ? = 5
[2,7,4,3,6,5,1] => [1,5,6,3,4,7,2] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [5,2,1,1,1,1]
 => ? = 4
[2,7,4,5,6,3,1] => [1,3,6,5,4,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,1,1]
 => ? = 4
[2,7,4,6,3,5,1] => [1,5,3,6,4,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,1,1,1]
 => ? = 4
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,1,1]
 => ? = 5
[2,7,5,3,6,4,1] => [1,4,6,3,5,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,2,1,1,1]
 => ? = 4
[2,7,5,4,6,3,1] => [1,3,6,4,5,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,1,1]
 => ? = 4
[2,7,5,6,3,4,1] => [1,4,3,6,5,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,3,1,1,1]
 => ? = 4
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,1,1]
 => ? = 5
[2,7,6,3,4,5,1] => [1,5,4,3,6,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,1,1,1]
 => ? = 4
[2,7,6,3,5,4,1] => [1,4,5,3,6,7,2] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,1,1]
 => ? = 5
[2,7,6,4,3,5,1] => [1,5,3,4,6,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [5,4,1,1,1,1]
 => ? = 4
[2,7,6,4,5,3,1] => [1,3,5,4,6,7,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,1,1]
 => ? = 5
[2,7,6,5,3,4,1] => [1,4,3,5,6,7,2] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,1,1]
 => ? = 5
[2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,1]
 => ? = 6
[3,2,6,7,5,4,1] => [1,4,5,7,6,2,3] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1,1,1]
 => ? = 4
[3,2,7,5,6,4,1] => [1,4,6,5,7,2,3] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,2,1,1,1]
 => ? = 4
[3,2,7,6,4,5,1] => [1,5,4,6,7,2,3] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,1,1,1]
 => ? = 4
[3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,1,1]
 => ? = 5
[3,4,5,6,7,2,1] => [1,2,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2,1]
 => ? = 3
[3,4,5,7,6,1,2] => [2,1,6,7,5,4,3] => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,2,2,2]
 => ? = 3
[3,4,5,7,6,2,1] => [1,2,6,7,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,2,2,2,1]
 => ? = 4
[3,4,6,5,7,2,1] => [1,2,7,5,6,4,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2,1]
 => ? = 3
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001068
(load all 13 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1,0]
 => 1
[1,2] => [2,1] => [1,1,0,0]
 => 1
[2,1] => [1,2] => [1,0,1,0]
 => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[6,5,7,8,3,4,1,2] => [2,1,4,3,8,7,5,6] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,4,3,5,6,7,1,2] => [2,1,7,6,5,3,4,8] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 3
[6,3,4,5,7,8,1,2] => [2,1,8,7,5,4,3,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[5,6,7,8,3,2,1,4] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[7,8,6,5,2,1,3,4] => [4,3,1,2,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[7,8,5,6,2,1,3,4] => [4,3,1,2,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[5,6,7,8,2,1,3,4] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
[6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[7,8,6,1,2,3,4,5] => [5,4,3,2,1,6,8,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[7,6,8,1,2,3,4,5] => [5,4,3,2,1,8,6,7] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[8,7,5,2,1,3,4,6] => [6,4,3,1,2,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
[7,8,5,2,1,3,4,6] => [6,4,3,1,2,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[8,7,5,1,2,3,4,6] => [6,4,3,2,1,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
[7,8,5,1,2,3,4,6] => [6,4,3,2,1,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[8,6,5,4,3,2,1,7] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
[8,4,5,6,3,2,1,7] => [7,1,2,3,6,5,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
[8,6,5,4,3,1,2,7] => [7,2,1,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
(load all 13 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1,0]
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5 - 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 6 - 1
[5,6,7,8,3,4,2,1] => [1,2,4,3,8,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 6 - 1
[5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 6 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[6,5,7,8,3,4,1,2] => [2,1,4,3,8,7,5,6] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[8,4,3,5,6,7,1,2] => [2,1,7,6,5,3,4,8] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 3 - 1
[6,3,4,5,7,8,1,2] => [2,1,8,7,5,4,3,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 1
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 1
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 - 1
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[5,6,7,8,4,2,1,3] => [3,1,2,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 - 1
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,3,2,1,4] => [4,1,2,3,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[7,8,6,5,2,1,3,4] => [4,3,1,2,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 1
[7,8,5,6,2,1,3,4] => [4,3,1,2,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[5,6,7,8,2,1,3,4] => [4,3,1,2,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5 - 1
[7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 1
[8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 - 1
[6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3 - 1
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
[7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 - 1
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 - 1
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 - 1
[7,8,6,1,2,3,4,5] => [5,4,3,2,1,6,8,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[7,6,8,1,2,3,4,5] => [5,4,3,2,1,8,6,7] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
[6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
[8,7,5,2,1,3,4,6] => [6,4,3,1,2,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[7,8,5,2,1,3,4,6] => [6,4,3,1,2,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
[8,7,5,1,2,3,4,6] => [6,4,3,2,1,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[7,8,5,1,2,3,4,6] => [6,4,3,2,1,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
[7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
[8,6,5,4,3,2,1,7] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[8,4,5,6,3,2,1,7] => [7,1,2,3,6,5,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[8,6,5,4,3,1,2,7] => [7,2,1,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
Description
The number of valleys of the Dyck path.
The following 66 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000105The number of blocks in the set partition. St000678The number of up steps after the last double rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000167The number of leaves of an ordered tree. St000996The number of exclusive left-to-right maxima of a permutation. St000912The number of maximal antichains in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000069The number of maximal elements of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000068The number of minimal elements in a poset. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000740The last entry of a permutation. St000654The first descent of a permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils 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. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000054The first entry of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000989The number of final rises of a permutation. St000990The first ascent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000015The number of peaks of a Dyck path. St000133The "bounce" of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000702The number of weak deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.