Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001392
(load all 3 compositions to match this statistic)
Mapping
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 2
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 3
[3,1]
 => 2
[2,2]
 => 1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 4
[4,1]
 => 3
[3,2]
 => 1
[3,1,1]
 => 2
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 5
[5,1]
 => 4
[4,2]
 => 3
[4,1,1]
 => 3
[3,3]
 => 2
[3,2,1]
 => 0
[3,1,1,1]
 => 2
[2,2,2]
 => 1
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 6
[6,1]
 => 5
[5,2]
 => 4
[5,1,1]
 => 4
[4,3]
 => 2
[4,2,1]
 => 3
[4,1,1,1]
 => 3
[3,3,1]
 => 2
[3,2,2]
 => 1
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 2
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 7
[7,1]
 => 6
[6,2]
 => 5
[6,1,1]
 => 5
[5,3]
 => 4
[5,2,1]
 => 4
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000733
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 59%
Values
[1]
 => [[1]]
 => [[1]]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [[1,2]]
 => [[1,2]]
 => [[1],[2]]
 => 2 = 1 + 1
[1,1]
 => [[1],[2]]
 => [[1],[2]]
 => [[1,2]]
 => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1 = 0 + 1
[1,1,1]
 => [[1],[2],[3]]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[3,1]
 => [[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 3 = 2 + 1
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 2 = 1 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1 = 0 + 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[5]
 => [[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
[4,1]
 => [[1,3,4,5],[2]]
 => [[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 4 = 3 + 1
[3,2]
 => [[1,2,5],[3,4]]
 => [[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => 2 = 1 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 3 = 2 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1 = 0 + 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => 1 = 0 + 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[6]
 => [[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => 5 = 4 + 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3,4,6],[2,5]]
 => [[1,2],[3,5],[4],[6]]
 => 4 = 3 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,3,5,6],[2],[4]]
 => [[1,2,4],[3],[5],[6]]
 => 4 = 3 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,3,5],[2,4,6]]
 => [[1,2],[3,4],[5,6]]
 => 3 = 2 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => [[1,3,6],[2,5],[4]]
 => 1 = 0 + 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,4,6],[2],[3],[5]]
 => [[1,2,3,5],[4],[6]]
 => 3 = 2 + 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,4],[2,5],[3,6]]
 => [[1,2,3],[4,5,6]]
 => 2 = 1 + 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,3],[2,5],[4],[6]]
 => [[1,2,4,6],[3,5]]
 => 1 = 0 + 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 1 = 0 + 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2,4,5,6,7],[3]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => 6 = 5 + 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3,4,6,7],[2,5]]
 => [[1,2],[3,5],[4],[6],[7]]
 => 5 = 4 + 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,3,5,6,7],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7]]
 => 5 = 4 + 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,3,5,6],[2,4,7]]
 => [[1,2],[3,4],[5,7],[6]]
 => 3 = 2 + 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4,7],[3,5],[6]]
 => [[1,3,6],[2,5],[4],[7]]
 => 4 = 3 + 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,4,6,7],[2],[3],[5]]
 => [[1,2,3,5],[4],[6],[7]]
 => 4 = 3 + 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6,7],[4]]
 => [[1,3,4],[2,6],[5,7]]
 => 3 = 2 + 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,4,6],[2,5],[3,7]]
 => [[1,2,3],[4,5,7],[6]]
 => 2 = 1 + 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,3,5],[2,6],[4],[7]]
 => [[1,2,4,7],[3,6],[5]]
 => 1 = 0 + 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,5,7],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5],[7]]
 => 3 = 2 + 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => 1 = 0 + 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,4],[2,6],[3],[5],[7]]
 => [[1,2,3,5,7],[4,6]]
 => 1 = 0 + 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,6],[2],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 1 = 0 + 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 8 = 7 + 1
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [[1,2,4,5,6,7,8],[3]]
 => [[1,3],[2],[4],[5],[6],[7],[8]]
 => 7 = 6 + 1
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3,4,6,7,8],[2,5]]
 => [[1,2],[3,5],[4],[6],[7],[8]]
 => 6 = 5 + 1
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [[1,3,5,6,7,8],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7],[8]]
 => 6 = 5 + 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [[1,3,5,6,8],[2,4,7]]
 => [[1,2],[3,4],[5,7],[6],[8]]
 => 5 = 4 + 1
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [[1,2,4,7,8],[3,5],[6]]
 => [[1,3,6],[2,5],[4],[7],[8]]
 => 5 = 4 + 1
[8,1]
 => [[1,3,4,5,6,7,8,9],[2]]
 => [[1,2,4,5,6,7,8,9],[3]]
 => [[1,3],[2],[4],[5],[6],[7],[8],[9]]
 => ? = 7 + 1
[7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => [[1,3,4,6,7,8,9],[2,5]]
 => [[1,2],[3,5],[4],[6],[7],[8],[9]]
 => ? = 6 + 1
[7,1,1]
 => [[1,4,5,6,7,8,9],[2],[3]]
 => [[1,3,5,6,7,8,9],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7],[8],[9]]
 => ? = 6 + 1
[6,3]
 => [[1,2,3,7,8,9],[4,5,6]]
 => [[1,3,5,6,8,9],[2,4,7]]
 => [[1,2],[3,4],[5,7],[6],[8],[9]]
 => ? = 5 + 1
[6,2,1]
 => [[1,3,6,7,8,9],[2,5],[4]]
 => [[1,2,4,7,8,9],[3,5],[6]]
 => [[1,3,6],[2,5],[4],[7],[8],[9]]
 => ? = 5 + 1
[6,1,1,1]
 => [[1,5,6,7,8,9],[2],[3],[4]]
 => [[1,4,6,7,8,9],[2],[3],[5]]
 => [[1,2,3,5],[4],[6],[7],[8],[9]]
 => ? = 5 + 1
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [[1,3,5,7,8],[2,4,6,9]]
 => [[1,2],[3,4],[5,6],[7,9],[8]]
 => ? = 3 + 1
[5,3,1]
 => [[1,3,4,8,9],[2,6,7],[5]]
 => [[1,2,5,7,9],[3,6,8],[4]]
 => [[1,3,4],[2,6],[5,8],[7],[9]]
 => ? = 4 + 1
[5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [[1,4,6,8,9],[2,5],[3,7]]
 => [[1,2,3],[4,5,7],[6],[8],[9]]
 => ? = 4 + 1
[5,2,1,1]
 => [[1,4,7,8,9],[2,6],[3],[5]]
 => [[1,3,5,8,9],[2,6],[4],[7]]
 => [[1,2,4,7],[3,6],[5],[8],[9]]
 => ? = 4 + 1
[5,1,1,1,1]
 => [[1,6,7,8,9],[2],[3],[4],[5]]
 => [[1,5,7,8,9],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5],[7],[8],[9]]
 => ? = 4 + 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [[1,2,5,6],[3,7,8,9],[4]]
 => [[1,3,4],[2,7],[5,8],[6,9]]
 => ? = 3 + 1
[4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => [[1,3,4,7],[2,6,8],[5,9]]
 => [[1,2,5],[3,6,9],[4,8],[7]]
 => ? = 1 + 1
[4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => [[1,2,6,9],[3,4],[5,7],[8]]
 => [[1,3,5,8],[2,4,7],[6],[9]]
 => ? = 3 + 1
[4,2,1,1,1]
 => [[1,5,8,9],[2,7],[3],[4],[6]]
 => [[1,4,6,9],[2,7],[3],[5],[8]]
 => [[1,2,3,5,8],[4,7],[6],[9]]
 => ? = 3 + 1
[4,1,1,1,1,1]
 => [[1,7,8,9],[2],[3],[4],[5],[6]]
 => [[1,6,8,9],[2],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6],[8],[9]]
 => ? = 3 + 1
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [[1,2,4],[3,5,7],[6,8],[9]]
 => [[1,3,6,9],[2,5,8],[4,7]]
 => ? = 0 + 1
[3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => [[1,4,7],[2,8,9],[3],[5],[6]]
 => [[1,2,3,5,6],[4,8],[7,9]]
 => ? = 2 + 1
[3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => [[1,5,8],[2,6],[3,7],[4,9]]
 => [[1,2,3,4],[5,6,7,9],[8]]
 => ? = 1 + 1
[3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => [[1,3,7],[2,5],[4,8],[6],[9]]
 => [[1,2,4,6,9],[3,5,8],[7]]
 => ? = 0 + 1
[3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => [[1,5,7],[2,8],[3],[4],[6],[9]]
 => [[1,2,3,4,6,9],[5,8],[7]]
 => ? = 0 + 1
[9,1]
 => [[1,3,4,5,6,7,8,9,10],[2]]
 => [[1,2,4,5,6,7,8,9,10],[3]]
 => [[1,3],[2],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 8 + 1
[8,2]
 => [[1,2,5,6,7,8,9,10],[3,4]]
 => [[1,3,4,6,7,8,9,10],[2,5]]
 => [[1,2],[3,5],[4],[6],[7],[8],[9],[10]]
 => ? = 7 + 1
[8,1,1]
 => [[1,4,5,6,7,8,9,10],[2],[3]]
 => [[1,3,5,6,7,8,9,10],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
 => ? = 7 + 1
[7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => [[1,3,5,6,8,9,10],[2,4,7]]
 => [[1,2],[3,4],[5,7],[6],[8],[9],[10]]
 => ? = 6 + 1
[7,2,1]
 => [[1,3,6,7,8,9,10],[2,5],[4]]
 => [[1,2,4,7,8,9,10],[3,5],[6]]
 => [[1,3,6],[2,5],[4],[7],[8],[9],[10]]
 => ? = 6 + 1
[7,1,1,1]
 => [[1,5,6,7,8,9,10],[2],[3],[4]]
 => [[1,4,6,7,8,9,10],[2],[3],[5]]
 => [[1,2,3,5],[4],[6],[7],[8],[9],[10]]
 => ? = 6 + 1
[6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => [[1,3,5,7,8,10],[2,4,6,9]]
 => [[1,2],[3,4],[5,6],[7,9],[8],[10]]
 => ? = 5 + 1
[6,3,1]
 => [[1,3,4,8,9,10],[2,6,7],[5]]
 => [[1,2,5,7,9,10],[3,6,8],[4]]
 => [[1,3,4],[2,6],[5,8],[7],[9],[10]]
 => ? = 5 + 1
[6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => [[1,4,6,8,9,10],[2,5],[3,7]]
 => [[1,2,3],[4,5,7],[6],[8],[9],[10]]
 => ? = 5 + 1
[6,2,1,1]
 => [[1,4,7,8,9,10],[2,6],[3],[5]]
 => [[1,3,5,8,9,10],[2,6],[4],[7]]
 => [[1,2,4,7],[3,6],[5],[8],[9],[10]]
 => ? = 5 + 1
[6,1,1,1,1]
 => [[1,6,7,8,9,10],[2],[3],[4],[5]]
 => [[1,5,7,8,9,10],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5],[7],[8],[9],[10]]
 => ? = 5 + 1
[5,4,1]
 => [[1,3,4,5,10],[2,7,8,9],[6]]
 => [[1,2,5,6,9],[3,7,8,10],[4]]
 => [[1,3,4],[2,7],[5,8],[6,10],[9]]
 => ? = 3 + 1
[5,3,2]
 => [[1,2,5,9,10],[3,4,8],[6,7]]
 => [[1,3,4,7,10],[2,6,8],[5,9]]
 => [[1,2,5],[3,6,9],[4,8],[7],[10]]
 => ? = 4 + 1
[5,2,2,1]
 => [[1,3,8,9,10],[2,5],[4,7],[6]]
 => [[1,2,6,9,10],[3,4],[5,7],[8]]
 => [[1,3,5,8],[2,4,7],[6],[9],[10]]
 => ? = 4 + 1
[5,2,1,1,1]
 => [[1,5,8,9,10],[2,7],[3],[4],[6]]
 => [[1,4,6,9,10],[2,7],[3],[5],[8]]
 => [[1,2,3,5,8],[4,7],[6],[9],[10]]
 => ? = 4 + 1
[5,1,1,1,1,1]
 => [[1,7,8,9,10],[2],[3],[4],[5],[6]]
 => [[1,6,8,9,10],[2],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6],[8],[9],[10]]
 => ? = 4 + 1
[4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => [[1,3,4,8],[2,6,9,10],[5,7]]
 => [[1,2,5],[3,6,7],[4,9],[8,10]]
 => ? = 3 + 1
[4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => [[1,3,6,7],[2,8,9,10],[4],[5]]
 => [[1,2,4,5],[3,8],[6,9],[7,10]]
 => ? = 3 + 1
[4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => [[1,4,7,9],[2,5,8],[3,6,10]]
 => [[1,2,3],[4,5,6],[7,8,10],[9]]
 => ? = 2 + 1
[4,3,1,1,1]
 => [[1,5,6,10],[2,8,9],[3],[4],[7]]
 => [[1,4,7,9],[2,8,10],[3],[5],[6]]
 => [[1,2,3,5,6],[4,8],[7,10],[9]]
 => ? = 2 + 1
[4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => [[1,5,8,10],[2,6],[3,7],[4,9]]
 => [[1,2,3,4],[5,6,7,9],[8],[10]]
 => ? = 3 + 1
[4,2,2,1,1]
 => [[1,4,9,10],[2,6],[3,8],[5],[7]]
 => [[1,3,7,10],[2,5],[4,8],[6],[9]]
 => [[1,2,4,6,9],[3,5,8],[7],[10]]
 => ? = 3 + 1
[4,2,1,1,1,1]
 => [[1,6,9,10],[2,8],[3],[4],[5],[7]]
 => [[1,5,7,10],[2,8],[3],[4],[6],[9]]
 => [[1,2,3,4,6,9],[5,8],[7],[10]]
 => ? = 3 + 1
[4,1,1,1,1,1,1]
 => [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
 => [[1,7,9,10],[2],[3],[4],[5],[6],[8]]
 => [[1,2,3,4,5,6,8],[7],[9],[10]]
 => ? = 3 + 1
[3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => [[1,2,5],[3,7,8],[4,9,10],[6]]
 => [[1,3,4,6],[2,7,9],[5,8,10]]
 => ? = 2 + 1
[3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6],[8,9]]
 => [[1,4,6],[2,5,9],[3,8],[7,10]]
 => [[1,2,3,7],[4,5,8,10],[6,9]]
 => ? = 1 + 1
[3,3,2,1,1]
 => [[1,4,7],[2,6,10],[3,9],[5],[8]]
 => [[1,3,5],[2,6,8],[4,9],[7],[10]]
 => [[1,2,4,7,10],[3,6,9],[5,8]]
 => ? = 0 + 1
[3,3,1,1,1,1]
 => [[1,6,7],[2,9,10],[3],[4],[5],[8]]
 => [[1,5,8],[2,9,10],[3],[4],[6],[7]]
 => [[1,2,3,4,6,7],[5,9],[8,10]]
 => ? = 2 + 1
[3,2,2,2,1]
 => [[1,3,10],[2,5],[4,7],[6,9],[8]]
 => [[1,2,8],[3,4],[5,6],[7,9],[10]]
 => [[1,3,5,7,10],[2,4,6,9],[8]]
 => ? = 0 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001463
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001463: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 0 + 2
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 3 = 1 + 2
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 2 + 2
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 5 = 3 + 2
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 2 + 2
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => 6 = 4 + 2
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 5 = 3 + 2
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 2 + 2
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 0 + 2
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 5 + 2
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 2
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 5 = 3 + 2
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 4 = 2 + 2
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? = 6 + 2
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5 + 2
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 2
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 2
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 2 + 2
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3 + 2
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 2 + 2
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 2
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[8]
 => 100000000 => [1,9] => ([(8,9)],10)
 => ? = 7 + 2
[7,1]
 => 100000010 => [1,7,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6 + 2
[6,2]
 => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5 + 2
[6,1,1]
 => 100000110 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5 + 2
[5,3]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 2
[5,2,1]
 => 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 2
[5,1,1,1]
 => 100001110 => [1,5,1,1,2] => ([(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 + 2
[4,4]
 => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 5 = 3 + 2
[4,3,1]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[4,2,2]
 => 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3 + 2
[4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3 + 2
[3,3,2]
 => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[3,1,1,1,1,1]
 => 100111110 => [1,3,1,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2 + 2
[2,2,2,2]
 => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 + 2
[2,2,1,1,1,1]
 => 11011110 => [1,1,2,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 + 2
[2,1,1,1,1,1,1]
 => 101111110 => [1,2,1,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0 + 2
[1,1,1,1,1,1,1,1]
 => 111111110 => [1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0 + 2
[9]
 => 1000000000 => [1,10] => ([(9,10)],11)
 => ? = 8 + 2
[8,1]
 => 1000000010 => [1,8,2] => ([(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 7 + 2
[7,2]
 => 100000100 => [1,6,3] => ([(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6 + 2
[7,1,1]
 => 1000000110 => [1,7,1,2] => ([(1,9),(1,10),(2,9),(2,10),(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 6 + 2
[6,3]
 => 10001000 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5 + 2
[6,2,1]
 => 100001010 => [1,5,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5 + 2
[6,1,1,1]
 => 1000001110 => [1,6,1,1,2] => ([(1,8),(1,9),(1,10),(2,8),(2,9),(2,10),(3,8),(3,9),(3,10),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 5 + 2
[5,4]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[5,3,1]
 => 10010010 => [1,3,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 2
[5,2,2]
 => 10001100 => [1,4,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 2
[5,2,1,1]
 => 100010110 => [1,4,2,1,2] => ([(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 + 2
[5,1,1,1,1]
 => 1000011110 => [1,5,1,1,1,2] => ([(1,7),(1,8),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 4 + 2
[4,4,1]
 => 1100010 => [1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 2
[4,3,2]
 => 1010100 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 + 2
[3,3,3]
 => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 2 + 2
Description
The number of distinct columns in the nullspace of a graph. Let $A$ be the adjacency matrix of a graph on $n$ vertices, and $K$ a $n\times d$ matrix whose column vectors form a basis of the nullspace of $A$. Then any other matrix $K'$ whose column vectors also form a basis of the nullspace is related to $K$ by $K' = K T$ for some invertible $d\times d$ matrix $T$. Any two rows of $K$ are equal if and only if they are equal in $K'$. The nullspace of a graph is usually written as a $d\times n$ matrix, hence the name of this statistic.
Matching statistic: St001232
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 35%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 5 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 3 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 3 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 4 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 4 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 0 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? = 7 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 6 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 5 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 4 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 4 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 3 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 1 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 2 + 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 3 + 1
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 2 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.