Your data matches 37 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000745
(load all 19 compositions to match this statistic)
Mapping
Mp00201: Dyck paths ā€”RingelāŸ¶ Permutations
Mp00059: Permutations ā€”Robinson-Schensted insertion tableauāŸ¶ Standard tableaux
St000745: Standard tableaux āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [[1],[2]]
 => 2
[1,0,1,0]
 => [3,1,2] => [[1,2],[3]]
 => 1
[1,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [[1,2],[3,4]]
 => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [[1,2],[3],[4]]
 => 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [[1,2,3,4],[5]]
 => 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [[1,2,3],[4,5]]
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [[1,2,4],[3,5]]
 => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [[1,2,3],[4],[5]]
 => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [[1,2,5],[3,4]]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [[1,3,4],[2,5]]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [[1,3,5],[2,4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [[1,2,4],[3],[5]]
 => 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [[1,2,3],[4],[5]]
 => 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [[1,2],[3,5],[4]]
 => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [[1,3,4],[2,5]]
 => 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [[1,3],[2,4],[5]]
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [[1,2],[3,4],[5]]
 => 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [[1,2,3,4],[5,6]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [[1,2,3,5],[4,6]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [[1,2,3,4],[5],[6]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [[1,2,3,6],[4,5]]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [[1,2,4,5],[3,6]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [[1,2,4],[3,5,6]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [[1,2,3,4],[5],[6]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [[1,2,3],[4,6],[5]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [[1,2,5],[3,4,6]]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [[1,2,4],[3,5],[6]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [[1,2,3],[4,5],[6]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [[1,2,5,6],[3,4]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [[1,3,4,5],[2,6]]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [[1,3,4],[2,5,6]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [[1,3,4],[2,5],[6]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [[1,3,5,6],[2,4]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [[1,2,4],[3,6],[5]]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [[1,2,3],[4,6],[5]]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [[1,2,5],[3,6],[4]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [[1,2,4],[3,5],[6]]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [[1,2,3],[4,5],[6]]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [[1,2,6],[3,5],[4]]
 => 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000297
(load all 9 compositions to match this statistic)
Mapping
Mp00119: Dyck paths ā€”to 321-avoiding permutation (Krattenthaler)āŸ¶ Permutations
Mp00109: Permutations ā€”descent wordāŸ¶ Binary words
St000297: Binary words āŸ¶ ā„¤Result quality: 99% ā—values known / values provided: 99%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => => ? = 2 - 1
[1,0,1,0]
 => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => 10 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0100 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1000 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,2,4,5,3,6,7,8] => ? => ? = 1 - 1
[]
 => [] => ? => ? = 1 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,1,5,6,7,9,2,3,8] => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 6 compositions to match this statistic)
Mapping
Mp00119: Dyck paths ā€”to 321-avoiding permutation (Krattenthaler)āŸ¶ Permutations
Mp00069: Permutations ā€”complementāŸ¶ Permutations
Mp00109: Permutations ā€”descent wordāŸ¶ Binary words
St000326: Binary words āŸ¶ ā„¤Result quality: 98% ā—values known / values provided: 98%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => => ? = 2
[1,0,1,0]
 => [1,2] => [2,1] => 1 => 1
[1,1,0,0]
 => [2,1] => [1,2] => 0 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 11 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 10 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,3,1] => 01 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,1,3] => 10 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 01 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 111 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,3,1,2] => 110 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 101 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,1,3] => 110 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [4,1,3,2] => 101 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,4,2,1] => 011 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 010 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,4,1] => 101 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,2,1,4] => 110 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => 101 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,4,3,1] => 011 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 010 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => 101 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,4,3,2] => 011 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => 1110 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,4,2,3,1] => 1101 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [5,4,2,1,3] => 1110 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,3,4,2,1] => 1011 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => 1101 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,4,2] => 1101 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,2,4,3,1] => 1011 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [5,2,4,1,3] => 1010 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [5,2,1,4,3] => 1101 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [5,1,4,3,2] => 1011 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,5,2,3,1] => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,5,2,1,3] => 0110 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,5,1,3,2] => 0101 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => 1011 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => 1010 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => 1101 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => 1101 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => 1101 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,6,2,3,4,5,8,7] => [8,3,7,6,5,4,1,2] => ? => ? = 1
[]
 => [] => [] => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,6,8,1,9,7] => [8,7,6,5,4,2,9,1,3] => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,4,5,7,8,1,9,6] => [8,7,6,5,3,2,9,1,4] => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,4,5,8,1,6,9,7] => [8,7,6,5,2,9,4,1,3] => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,8,9,1,6,7] => [8,7,6,5,2,1,9,4,3] => ? => ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,3,4,7,8,9,1,5,6] => [8,7,6,3,2,1,9,5,4] => ? => ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [2,3,6,7,8,9,1,4,5] => [8,7,4,3,2,1,9,6,5] => ? => ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,1,5,6,7,9,2,3,8] => [6,9,5,4,3,1,8,7,2] => ? => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1,9,6,7,8] => [8,7,6,5,9,1,4,3,2] => ? => ? = 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000877
Mapping
Mp00119: Dyck paths ā€”to 321-avoiding permutation (Krattenthaler)āŸ¶ Permutations
Mp00069: Permutations ā€”complementāŸ¶ Permutations
Mp00109: Permutations ā€”descent wordāŸ¶ Binary words
St000877: Binary words āŸ¶ ā„¤Result quality: 98% ā—values known / values provided: 98%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => => ? = 2 - 1
[1,0,1,0]
 => [1,2] => [2,1] => 1 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [1,2] => 0 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 11 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 10 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,3,1] => 01 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1,3] => 10 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,3,1,2] => 110 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,1,3] => 110 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [4,1,3,2] => 101 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,4,2,1] => 011 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,4,1] => 101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,2,1,4] => 110 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => 101 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,4,3,1] => 011 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 010 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => 101 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,4,3,2] => 011 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => 1110 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,4,2,3,1] => 1101 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [5,4,2,1,3] => 1110 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,3,4,2,1] => 1011 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => 1101 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,4,2] => 1101 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,2,4,3,1] => 1011 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [5,2,4,1,3] => 1010 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [5,2,1,4,3] => 1101 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [5,1,4,3,2] => 1011 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,5,2,3,1] => 0101 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,5,2,1,3] => 0110 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,5,1,3,2] => 0101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => 1011 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => 1010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => 1101 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => 1101 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => 1101 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 1 = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,6,2,3,4,5,8,7] => [8,3,7,6,5,4,1,2] => ? => ? = 1 - 1
[]
 => [] => [] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,6,8,1,9,7] => [8,7,6,5,4,2,9,1,3] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,4,5,7,8,1,9,6] => [8,7,6,5,3,2,9,1,4] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,4,5,8,1,6,9,7] => [8,7,6,5,2,9,4,1,3] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,8,9,1,6,7] => [8,7,6,5,2,1,9,4,3] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,3,4,7,8,9,1,5,6] => [8,7,6,3,2,1,9,5,4] => ? => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [2,3,6,7,8,9,1,4,5] => [8,7,4,3,2,1,9,6,5] => ? => ? = 1 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,1,5,6,7,9,2,3,8] => [6,9,5,4,3,1,8,7,2] => ? => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1,9,6,7,8] => [8,7,6,5,9,1,4,3,2] => ? => ? = 1 - 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000932
(load all 2 compositions to match this statistic)
Mapping
Mp00119: Dyck paths ā€”to 321-avoiding permutation (Krattenthaler)āŸ¶ Permutations
Mp00071: Permutations ā€”descent compositionāŸ¶ Integer compositions
Mp00231: Integer compositions ā€”bounce pathāŸ¶ Dyck paths
St000932: Dyck paths āŸ¶ ā„¤Result quality: 89% ā—values known / values provided: 89%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 2 - 1
[1,0,1,0]
 => [1,2] => [2] => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[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] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,5,7,6,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,4,6,7,5,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,5,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,5,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,5,6,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,4,8,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,2,4,5,3,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,6,8,3,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,2,6,7,8,3,4,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,2,7,8,3,4,5,6] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,4,5,6,7,8,2,3] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,6,7,8,2,3,4,5] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,4,1,5,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,4,5,1,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,5,6,1,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,6,7,1,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,7,1,8,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,7,8,1,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,4,6,7,8,1,5] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,4,7,8,1,5,6] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [2,3,5,6,1,4,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,3,6,7,8,1,4,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [2,4,6,7,8,1,3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,8,1,3,4] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,6,7,8,1,3,4,5] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,6,7,1,8,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,8,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,8,1,2,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9,2,3,4,5,6,7,8] => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 - 1
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,9,3,4,5,6,7,8] => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 1
[]
 => [] => [] => ?
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,1] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,9,1,8] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,6,8,1,9,7] => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,6,8,9,1,7] => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => ? = 1 - 1
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000504
(load all 2 compositions to match this statistic)
Mapping
Mp00035: Dyck paths ā€”to alternating sign matrixāŸ¶ Alternating sign matrices
Mp00002: Alternating sign matrices ā€”to left key permutationāŸ¶ Permutations
Mp00240: Permutations ā€”weak exceedance partitionāŸ¶ Set partitions
St000504: Set partitions āŸ¶ ā„¤Result quality: 86% ā—values known / values provided: 86%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => {{1}}
 => ? = 2
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => {{1},{2}}
 => 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => {{1,2}}
 => 2
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => {{1},{2},{3}}
 => 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => {{1,2},{3}}
 => 2
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => {{1},{2,3}}
 => 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => {{1,3},{2}}
 => 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
 => ? => ?
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,4,6,5,7,8] => ?
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,6,5,8,7] => ?
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => ? => ?
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,6,5,8,7] => ?
 => ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,8},{5},{6},{7}}
 => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => ? => ?
 => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,-1,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,8},{5},{6},{7}}
 => ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,8,3,4,5,6,7] => ?
 => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => ? => ?
 => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
 => [1,6,2,3,4,5,8,7] => ?
 => ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,-1,1,0],[0,0,1,0,0,0,-1,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,8},{5},{6},{7}}
 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
 => ? = 2
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [2,1,3,8,4,5,6,7] => {{1,2},{3},{4,8},{5},{6},{7}}
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
 => ? => ?
 => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [2,1,8,3,4,5,6,7] => {{1,2},{3,8},{4},{5},{6},{7}}
 => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
 => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
 => [1,3,2,4,5,6,8,7] => ?
 => ? = 1
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,3,2,4,8,5,6,7] => {{1},{2,3},{4},{5,8},{6},{7}}
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,2,4,3,5,6,7,8] => ?
 => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,4,6,5,7,8] => ?
 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,6,5,8,7] => ?
 => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => ? => ?
 => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
 => [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,6},{5},{7},{8}}
 => ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,0,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,-1,0,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,0,0,1,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 1
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [[0,1,0,0,0,0,0,0],[1,-1,0,0,0,1,0,0],[0,1,0,0,0,-1,1,0],[0,0,1,0,0,0,-1,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
 => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,8},{5},{6},{7}}
 => ? = 1
Description
The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St001217
(load all 70 compositions to match this statistic)
Mapping
Mp00027: Dyck paths ā€”to partitionāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
St001217: Dyck paths āŸ¶ ā„¤Result quality: 85% ā—values known / values provided: 85%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,0,1,0]
 => [1]
 => []
 => []
 => ? = 1 - 1
[1,1,0,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2]
 => []
 => []
 => ? = 2 - 1
[1,1,0,1,0,0]
 => [1]
 => []
 => []
 => ? = 1 - 1
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 => []
 => ? = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 => []
 => ? = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 => []
 => ? = 1 - 1
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => []
 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 => []
 => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [5]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [4]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2]
 => []
 => []
 => ? = 2 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => []
 => []
 => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,2,1]
 => [6,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2,1]
 => [5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,5,4,3,2,1]
 => [5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3,2,1]
 => [5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => [6,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,4,3,2,1]
 => [5,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,4,3,2,1]
 => [5,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,4,4,3,2,1]
 => [4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,4,4,3,2,1]
 => [4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3,2,1]
 => [4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ?
 => ?
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,3,2,1]
 => [5,4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,3,3,2,1]
 => [4,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,2,1]
 => [3,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,2,2,1]
 => ?
 => ?
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,2,1]
 => [5,4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,5,4,3,2,2,1]
 => [5,4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,2,2,2,1]
 => [3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,1]
 => [6,5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000390
Mapping
Mp00201: Dyck paths ā€”RingelāŸ¶ Permutations
Mp00089: Permutations ā€”Inverse Kreweras complementāŸ¶ Permutations
Mp00114: Permutations ā€”connectivity setāŸ¶ Binary words
St000390: Binary words āŸ¶ ā„¤Result quality: 79% ā—values known / values provided: 79%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => 1 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => 00 => 0 = 1 - 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => 11 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 000 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,1,3,2] => 000 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,4,2,3] => 100 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,1,3] => 000 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 111 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => 0000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,5,1,4,2] => 0000 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [4,1,5,3,2] => 0000 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,5,3,1,2] => 0000 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => 0000 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,4,5,2,3] => 1000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => 1000 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,2,5,1,3] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,2,1,3] => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,1,4,3] => 0000 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,5,3,4] => 1100 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,5,3,2,4] => 1000 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,1,4] => 0000 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,6,1,5,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,5,1,6,4,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,5,6,4,1,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,6,1,4,5,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,1,5,6,3,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,1,6,3,5,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,5,3,6,1,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,5,6,3,1,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,6,3,1,5,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [5,1,3,6,4,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,1,6,4,3,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,6,3,4,1,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,1,3,4,5,2] => 00000 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,4,5,6,2,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,4,6,2,5,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,5,2,6,4,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,5,6,4,2,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,6,2,4,5,3] => 10000 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,2,5,6,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,2,6,1,5,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,2,6,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,6,2,1,5,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [5,2,1,6,4,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [5,2,6,4,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,6,2,4,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,1,4,5,3] => 00000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,1,2,3,4,8,5,7] => [3,4,5,7,1,8,6,2] => ? => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,1,2,3,8,4,6,7] => [3,4,6,1,7,8,5,2] => ? => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,1,2,3,7,4,8,6] => [3,4,6,1,8,5,7,2] => ? => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [8,1,2,3,6,4,5,7] => [3,4,6,7,5,8,1,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,1,2,7,3,5,8,6] => [3,5,1,6,8,4,7,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,1,2,6,3,7,8,5] => [3,5,1,8,4,6,7,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [3,5,6,4,8,1,7,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [3,5,6,7,4,8,1,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,1,2,5,3,8,4,7] => [3,5,7,4,1,8,6,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => [3,5,7,8,4,6,1,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,1,2,5,7,3,8,6] => [3,6,1,4,8,5,7,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,8,6,3,5,7] => [3,6,1,7,5,8,4,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,5,6,8,3,7] => [3,7,1,4,5,8,6,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,8,2,4,5,6,7] => [4,1,5,6,7,8,3,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [3,1,7,2,4,5,8,6] => [4,1,5,6,8,3,7,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [3,1,8,2,4,7,5,6] => [4,1,5,7,8,6,3,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [3,1,8,2,6,4,5,7] => [4,1,6,7,5,8,3,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,1,5,2,8,7,4,6] => [4,1,7,3,8,6,5,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,1,8,2,6,7,4,5] => [4,1,7,8,5,6,3,2] => ? => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,1,5,2,6,7,8,4] => [4,1,8,3,5,6,7,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,1,4,2,3,8,5,7] => [4,5,3,7,1,8,6,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [6,1,4,2,3,7,8,5] => [4,5,3,8,1,6,7,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [4,5,6,3,8,1,7,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => [4,5,7,3,8,6,1,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [8,1,4,2,7,3,5,6] => [4,6,3,7,8,5,1,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => [4,7,8,3,5,6,1,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,4,8,2,5,6,7] => [5,1,3,6,7,8,4,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,4,7,2,5,8,6] => [5,1,3,6,8,4,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,1,4,6,2,8,5,7] => [5,1,3,7,4,8,6,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [3,1,8,5,2,4,6,7] => [5,1,6,4,7,8,3,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [3,1,7,5,2,4,8,6] => [5,1,6,4,8,3,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,7,6,2,4,8,5] => [5,1,6,8,4,3,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,8,6,2,7,4,5] => [5,1,7,8,4,6,3,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,5,2,7,8,4] => [5,1,8,4,3,6,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [7,1,4,5,2,3,8,6] => [5,6,3,4,8,1,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,1,4,5,2,8,3,7] => [5,7,3,4,1,8,6,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [3,1,4,5,8,2,6,7] => [6,1,3,4,7,8,5,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [3,1,4,5,7,2,8,6] => [6,1,3,4,8,5,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,1,4,7,6,2,8,5] => [6,1,3,8,5,4,7,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [3,1,8,5,6,2,4,7] => [6,1,7,4,5,8,3,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [8,1,4,5,6,2,3,7] => [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,4,5,6,8,2,7] => [7,1,3,4,5,8,6,2] => ? => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,6,1,3,4,8,5,7] => [1,4,5,7,2,8,6,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,8,1,3,4,7,5,6] => [1,4,5,7,8,6,2,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,6,1,3,4,7,8,5] => [1,4,5,8,2,6,7,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [2,8,1,3,6,4,5,7] => [1,4,6,7,5,8,2,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,8,1,3,7,4,5,6] => [1,4,6,7,8,5,2,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [2,7,1,3,6,4,8,5] => [1,4,6,8,5,2,7,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [2,5,1,3,8,7,4,6] => [1,4,7,2,8,6,5,3] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,5,1,3,6,7,8,4] => [1,4,8,2,5,6,7,3] => ? => ? = 2 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000541
(load all 5 compositions to match this statistic)
Mapping
Mp00035: Dyck paths ā€”to alternating sign matrixāŸ¶ Alternating sign matrices
Mp00002: Alternating sign matrices ā€”to left key permutationāŸ¶ Permutations
Mp00066: Permutations ā€”inverseāŸ¶ Permutations
St000541: Permutations āŸ¶ ā„¤Result quality: 68% ā—values known / values provided: 68%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => [1] => ? = 2 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => [2,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,6,4,5,7] => [2,1,3,5,6,4,7] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,6,4,5,7] => [2,1,3,5,6,4,7] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2 - 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,5,3,4,6,7] => [2,1,4,5,3,6,7] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,6,4,5,7] => [2,1,3,5,6,4,7] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,6,3,4,5,7] => [2,1,4,5,6,3,7] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,7,3,4,5,6] => [2,1,4,5,6,7,3] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [3,1,2,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [3,1,2,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [3,1,2,4,7,5,6] => [2,3,1,4,6,7,5] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [3,1,2,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [3,1,2,5,4,7,6] => [2,3,1,5,4,7,6] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
 => [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [3,1,2,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000990
(load all 58 compositions to match this statistic)
Mapping
Mp00035: Dyck paths ā€”to alternating sign matrixāŸ¶ Alternating sign matrices
Mp00002: Alternating sign matrices ā€”to left key permutationāŸ¶ Permutations
St000990: Permutations āŸ¶ ā„¤Result quality: 65% ā—values known / values provided: 65%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => ? = 2
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 2
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 2
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [1,5,2,3,4,6,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [1,5,2,3,4,7,6] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [1,6,2,3,4,5,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [1,5,2,3,4,7,6] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [1,7,2,3,4,5,6] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,5,6,7] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,3,5,4,6,7] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,6,4,5,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,5,6,7] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,3,5,4,6,7] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,6,4,5,7] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [2,1,5,3,4,6,7] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,4,3,6,5,7] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,4,3,5,7,6] => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,3,6,4,5,7] => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,3,5,4,7,6] => ? = 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,4,7,5,6] => ? = 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [2,1,6,3,4,5,7] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [2,1,5,3,4,7,6] => ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,4,3,7,5,6] => ? = 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,3,7,4,5,6] => ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [2,1,7,3,4,5,6] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => [1,5,2,3,4,7,6] => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [1,6,2,3,4,5,7] => ? = 1
Description
The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000678The number of up steps after the last double rise of a Dyck path. St000352The Elizalde-Pak rank of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000237The number of small exceedances. St000989The number of final rises of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000654The first descent of a permutation. St000061The number of nodes on the left branch of a binary tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St000234The number of global ascents of a permutation. St000542The number of left-to-right-minima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001545The second Elser number of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001816Eigenvalues of the top-to-random operator acting on a simple module.