Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000681
Mp00083: Standard tableaux shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001514
Mp00083: Standard tableaux shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00122: Dyck paths Elizalde-Deutsch bijectionDyck paths
St001514: Dyck paths ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 38%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,4,6],[3],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,3,6],[4],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,3,4,5],[2],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,4,5],[3],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,3,5],[4],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,6],[3,4],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,5],[2,4],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,5],[3,4],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,5,7],[2,6],[3],[4]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,4,7],[2,6],[3],[5]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,3,7],[2,6],[4],[5]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,2,7],[3,6],[4],[5]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,4,7],[2,5],[3],[6]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,3,7],[2,5],[4],[6]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,2,7],[3,5],[4],[6]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,3,7],[2,4],[5],[6]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,2,7],[3,4],[5],[6]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,5,6],[2,7],[3],[4]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,4,6],[2,7],[3],[5]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,3,6],[2,7],[4],[5]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,2,6],[3,7],[4],[5]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,4,5],[2,7],[3],[6]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[[1,5],[2,6],[3,7],[4]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,4],[2,6],[3,7],[5]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,3],[2,6],[4,7],[5]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,2],[3,6],[4,7],[5]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,4],[2,5],[3,7],[6]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,3],[2,5],[4,7],[6]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,2],[3,5],[4,7],[6]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,3],[2,4],[5,7],[6]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[[1,2],[3,4],[5,7],[6]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Mp00083: Standard tableaux shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 25%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 2
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,4,6],[3],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,3,6],[4],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,3,4,5],[2],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,4,5],[3],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,2,3,5],[4],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,6],[3,4],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,5],[2,4],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,5],[3,4],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,4,7,8],[2,5],[3,6]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,7,8],[2,5],[4,6]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,7,8],[3,5],[4,6]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,7,8],[2,4],[5,6]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,7,8],[3,4],[5,6]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,4,6,8],[2,5],[3,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,6,8],[2,5],[4,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,6,8],[3,5],[4,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,6,8],[2,4],[5,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,6,8],[3,4],[5,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,4,5,8],[2,6],[3,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,5,8],[2,6],[4,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,5,8],[3,6],[4,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,4,8],[2,6],[5,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,4,8],[3,6],[5,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,3,8],[4,6],[5,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,5,8],[2,4],[6,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,5,8],[3,4],[6,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,4,8],[2,5],[6,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,4,8],[3,5],[6,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,3,8],[4,5],[6,7]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,4,6,7],[2,5],[3,8]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,3,6,7],[2,5],[4,8]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
[[1,2,6,7],[3,5],[4,8]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 1 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Mp00081: Standard tableaux reading word permutationPermutations
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00069: Permutations complementPermutations
St001682: Permutations ⟶ ℤResult quality: 8% values known / values provided: 8%distinct values known / distinct values provided: 50%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 3 = 2 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [3,4,5,6,2,1] => 2 = 1 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4,6] => [2,4,5,6,3,1] => 2 = 1 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,4,2,1,3,6] => [2,3,5,6,4,1] => 2 = 1 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [5,4,3,1,2,6] => [2,3,4,6,5,1] => 2 = 1 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,4,2,1,3,5] => [1,3,5,6,4,2] => 2 = 1 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [6,4,3,1,2,5] => [1,3,4,6,5,2] => 2 = 1 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => 2 = 1 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [6,5,3,1,2,4] => [1,2,4,6,5,3] => 2 = 1 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => 2 = 1 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,2,6,1,5] => [3,4,5,1,6,2] => 2 = 1 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [4,3,2,6,1,5] => [3,4,5,1,6,2] => 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [4,3,2,6,1,5] => [3,4,5,1,6,2] => 2 = 1 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [4,3,2,6,1,5] => [3,4,5,1,6,2] => 2 = 1 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [6,3,2,5,1,4] => [1,4,5,2,6,3] => 2 = 1 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [6,3,2,5,1,4] => [1,4,5,2,6,3] => 2 = 1 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [6,3,2,5,1,4] => [1,4,5,2,6,3] => 2 = 1 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,5,2,4,1,3] => [1,2,5,3,6,4] => 2 = 1 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [6,5,2,4,1,3] => [1,2,5,3,6,4] => 2 = 1 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [2,3,4,5,6,1] => 3 = 2 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [6,4,3,2,1,5] => [1,3,4,5,6,2] => 3 = 2 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [6,5,3,2,1,4] => [1,2,4,5,6,3] => 3 = 2 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [6,5,4,2,1,3] => [1,2,3,5,6,4] => 3 = 2 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 3 = 2 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 4 = 3 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => ? = 1 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [5,3,2,1,4,6,7] => [3,5,6,7,4,2,1] => ? = 1 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [5,4,2,1,3,6,7] => [3,4,6,7,5,2,1] => ? = 1 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [5,4,3,1,2,6,7] => [3,4,5,7,6,2,1] => ? = 1 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [6,3,2,1,4,5,7] => [2,5,6,7,4,3,1] => ? = 1 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => ? = 1 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [6,4,3,1,2,5,7] => [2,4,5,7,6,3,1] => ? = 1 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [6,5,2,1,3,4,7] => [2,3,6,7,5,4,1] => ? = 1 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [6,5,3,1,2,4,7] => [2,3,5,7,6,4,1] => ? = 1 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [6,5,4,1,2,3,7] => [2,3,4,7,6,5,1] => ? = 1 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [1,5,6,7,4,3,2] => ? = 1 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [1,4,6,7,5,3,2] => ? = 1 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => 2 = 1 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [7,5,2,1,3,4,6] => [1,3,6,7,5,4,2] => 2 = 1 + 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => 2 = 1 + 1
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 2 = 1 + 1
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [7,6,2,1,3,4,5] => [1,2,6,7,5,4,3] => 2 = 1 + 1
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [7,6,3,1,2,4,5] => [1,2,5,7,6,4,3] => 2 = 1 + 1
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [7,6,4,1,2,3,5] => [1,2,4,7,6,5,3] => 2 = 1 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 2 = 1 + 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,6,2,5,1,4,7] => [5,2,6,3,7,4,1] => ? = 1 + 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [3,6,2,5,1,4,7] => [5,2,6,3,7,4,1] => ? = 1 + 1
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [3,6,2,5,1,4,7] => [5,2,6,3,7,4,1] => ? = 1 + 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [3,6,2,5,1,4,7] => [5,2,6,3,7,4,1] => ? = 1 + 1
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [3,6,2,5,1,4,7] => [5,2,6,3,7,4,1] => ? = 1 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,7,2,5,1,4,6] => [5,1,6,3,7,4,2] => ? = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [3,7,2,5,1,4,6] => [5,1,6,3,7,4,2] => ? = 1 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [3,7,2,5,1,4,6] => [5,1,6,3,7,4,2] => ? = 1 + 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [5,7,2,4,1,3,6] => [3,1,6,4,7,5,2] => ? = 1 + 1
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [5,7,2,4,1,3,6] => [3,1,6,4,7,5,2] => ? = 1 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,7,2,6,1,4,5] => [5,1,6,2,7,4,3] => ? = 1 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [4,7,2,6,1,3,5] => [4,1,6,2,7,5,3] => ? = 1 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [4,7,3,6,1,2,5] => [4,1,5,2,7,6,3] => ? = 1 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [4,7,2,6,1,3,5] => [4,1,6,2,7,5,3] => ? = 1 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [4,7,3,6,1,2,5] => [4,1,5,2,7,6,3] => ? = 1 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [4,7,3,6,1,2,5] => [4,1,5,2,7,6,3] => ? = 1 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [5,7,2,4,1,3,6] => [3,1,6,4,7,5,2] => ? = 1 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [5,7,2,4,1,3,6] => [3,1,6,4,7,5,2] => ? = 1 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [4,7,2,6,1,3,5] => [4,1,6,2,7,5,3] => ? = 1 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [4,7,3,6,1,2,5] => [4,1,5,2,7,6,3] => ? = 1 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [4,7,3,6,1,2,5] => [4,1,5,2,7,6,3] => ? = 1 + 1
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,3,2,6,1,5,7] => [4,5,6,2,7,3,1] => ? = 1 + 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [4,3,2,6,1,5,7] => [4,5,6,2,7,3,1] => ? = 1 + 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [4,3,2,6,1,5,7] => [4,5,6,2,7,3,1] => ? = 1 + 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [4,3,2,6,1,5,7] => [4,5,6,2,7,3,1] => ? = 1 + 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [6,3,2,5,1,4,7] => [2,5,6,3,7,4,1] => ? = 1 + 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [6,3,2,5,1,4,7] => [2,5,6,3,7,4,1] => ? = 1 + 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [6,3,2,5,1,4,7] => [2,5,6,3,7,4,1] => ? = 1 + 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [6,5,2,4,1,3,7] => [2,3,6,4,7,5,1] => ? = 1 + 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [6,5,2,4,1,3,7] => [2,3,6,4,7,5,1] => ? = 1 + 1
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [4,3,2,7,1,5,6] => [4,5,6,1,7,3,2] => ? = 1 + 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [5,3,2,7,1,4,6] => [3,5,6,1,7,4,2] => ? = 1 + 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [5,4,2,7,1,3,6] => [3,4,6,1,7,5,2] => ? = 1 + 1
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [5,4,3,7,1,2,6] => [3,4,5,1,7,6,2] => ? = 1 + 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [5,3,2,7,1,4,6] => [3,5,6,1,7,4,2] => ? = 1 + 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [5,4,2,7,1,3,6] => [3,4,6,1,7,5,2] => ? = 1 + 1
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => [5,4,3,7,1,2,6] => [3,4,5,1,7,6,2] => ? = 1 + 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [5,4,2,7,1,3,6] => [3,4,6,1,7,5,2] => ? = 1 + 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [7,5,2,4,1,3,6] => [1,3,6,4,7,5,2] => 2 = 1 + 1
[[1,2,6],[3,4],[5],[7]]
=> [7,5,3,4,1,2,6] => [7,5,2,4,1,3,6] => [1,3,6,4,7,5,2] => 2 = 1 + 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [7,4,2,6,1,3,5] => [1,4,6,2,7,5,3] => 2 = 1 + 1
[[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => [7,4,3,6,1,2,5] => [1,4,5,2,7,6,3] => 2 = 1 + 1
[[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => [7,4,2,6,1,3,5] => [1,4,6,2,7,5,3] => 2 = 1 + 1
[[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => [7,4,3,6,1,2,5] => [1,4,5,2,7,6,3] => 2 = 1 + 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.
Matching statistic: St000528
Mp00083: Standard tableaux shapeInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
St000528: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 50%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 1 + 3
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4 = 1 + 3
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 4 = 1 + 3
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 1 + 3
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 4 = 1 + 3
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5 = 2 + 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 3 + 3
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 4 = 1 + 3
[[1,4,7],[2,5],[3,6]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,7],[2,5],[4,6]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,7],[3,5],[4,6]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,7],[2,4],[5,6]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,7],[3,4],[5,6]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,4,6],[2,5],[3,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,6],[2,5],[4,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,6],[3,5],[4,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,6],[2,4],[5,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,6],[3,4],[5,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,4,5],[2,6],[3,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,5],[2,6],[4,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,5],[3,6],[4,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,4],[2,6],[5,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,4],[3,6],[5,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,3],[4,6],[5,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,5],[2,4],[6,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,5],[3,4],[6,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,3,4],[2,5],[6,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,4],[3,5],[6,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,2,3],[4,5],[6,7]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 3
[[1,5,7],[2,6],[3],[4]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,4,7],[2,6],[3],[5]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,7],[2,6],[4],[5]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,7],[3,6],[4],[5]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,4,7],[2,5],[3],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,7],[2,5],[4],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,7],[3,5],[4],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,7],[2,4],[5],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,7],[3,4],[5],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,5,6],[2,7],[3],[4]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,4,6],[2,7],[3],[5]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,6],[2,7],[4],[5]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,6],[3,7],[4],[5]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,4,5],[2,7],[3],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,5],[2,7],[4],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,5],[3,7],[4],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,4],[2,7],[5],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,4],[3,7],[5],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,3],[4,7],[5],[6]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,4,6],[2,5],[3],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,6],[2,5],[4],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,6],[3,5],[4],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,6],[2,4],[5],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,6],[3,4],[5],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,4,5],[2,6],[3],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,5],[2,6],[4],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,5],[3,6],[4],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,3,4],[2,6],[5],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
[[1,2,4],[3,6],[5],[7]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 1 + 3
Description
The height of a poset. This equals the rank of the poset [[St000080]] plus one.
Mp00081: Standard tableaux reading word permutationPermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000454: Graphs ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 50%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [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)
=> 4 = 2 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [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)
=> 4 = 2 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [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)
=> 4 = 2 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [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)
=> 4 = 2 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [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)
=> 4 = 2 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [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)
=> 4 = 2 + 2
[[1,5,7],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7] => [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)
=> 4 = 2 + 2
[[1,4,7],[2],[3],[5],[6]]
=> [6,5,3,2,1,4,7] => [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)
=> 4 = 2 + 2
[[1,3,7],[2],[4],[5],[6]]
=> [6,5,4,2,1,3,7] => [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)
=> 4 = 2 + 2
[[1,2,7],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7] => [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)
=> 4 = 2 + 2
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [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)
=> 4 = 2 + 2
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [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)
=> 4 = 2 + 2
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [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)
=> 4 = 2 + 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000007
Mp00081: Standard tableaux reading word permutationPermutations
Mp00126: Permutations cactus evacuationPermutations
Mp00088: Permutations Kreweras complementPermutations
St000007: Permutations ⟶ ℤResult quality: 5% values known / values provided: 5%distinct values known / distinct values provided: 88%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 1 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [1,5,4,2,3] => 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [1,5,2,4,3] => 3 = 1 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [1,2,5,4,3] => 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [2,1,5,4,3] => 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 2 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,5,6,3,2,1] => [1,6,5,2,3,4] => 3 = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,5,6,4,2,1] => [1,6,2,5,3,4] => 3 = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [2,5,6,4,3,1] => [1,2,6,5,3,4] => 3 = 1 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,5,6,4,3,2] => [2,1,6,5,3,4] => 3 = 1 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,4,6,5,2,1] => [1,6,2,3,5,4] => 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,5,3,1] => [1,2,6,3,5,4] => 3 = 1 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,4,6,5,3,2] => [2,1,6,3,5,4] => 3 = 1 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [2,3,6,5,4,1] => [1,2,3,6,5,4] => 3 = 1 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,3,6,5,4,2] => [2,1,3,6,5,4] => 3 = 1 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => [2,3,1,6,5,4] => 3 = 1 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,2,5,1,4] => [6,4,2,1,5,3] => 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,5,1,3] => [6,4,1,2,5,3] => 3 = 1 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [4,6,3,5,1,2] => [6,1,4,2,5,3] => 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,6,2,4,1,3] => [6,4,1,5,2,3] => 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [6,1,4,5,2,3] => 3 = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,6,3,5,2,1] => [1,6,4,2,5,3] => 3 = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,6,3,4,2,1] => [1,6,4,5,2,3] => 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [5,6,2,4,3,1] => [1,4,6,5,2,3] => 3 = 1 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,6,1,4,3,2] => [4,1,6,5,2,3] => 3 = 1 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,5,4,1] => [1,4,2,6,5,3] => 3 = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,6,2,5,3,1] => [1,4,6,2,5,3] => 3 = 1 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,6,1,5,3,2] => [4,1,6,2,5,3] => 3 = 1 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [2,6,1,5,4,3] => [4,2,1,6,5,3] => 3 = 1 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,6,1,5,4,2] => [4,1,2,6,5,3] => 3 = 1 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,6,4,3,2,1] => [1,6,5,4,2,3] => 4 = 2 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,5,3,2,1] => [1,6,5,2,4,3] => 4 = 2 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,6,5,4,2,1] => [1,6,2,5,4,3] => 4 = 2 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [2,6,5,4,3,1] => [1,2,6,5,4,3] => 4 = 2 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => [2,1,6,5,4,3] => 4 = 2 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,6,5,4,3,2] => 5 = 3 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => [1,7,6,2,3,4,5] => ? = 1 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,6,7,4,2,1] => [1,7,2,6,3,4,5] => ? = 1 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,6,7,4,3,1] => [1,2,7,6,3,4,5] => ? = 1 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,5,6,7,4,3,2] => [2,1,7,6,3,4,5] => ? = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [3,4,6,7,5,2,1] => [1,7,2,3,6,4,5] => ? = 1 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => [1,2,7,3,6,4,5] => ? = 1 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,4,6,7,5,3,2] => [2,1,7,3,6,4,5] => ? = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [2,3,6,7,5,4,1] => [1,2,3,7,6,4,5] => ? = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,3,6,7,5,4,2] => [2,1,3,7,6,4,5] => ? = 1 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,2,6,7,5,4,3] => [2,3,1,7,6,4,5] => ? = 1 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [3,4,5,7,6,2,1] => [1,7,2,3,4,6,5] => ? = 1 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,5,7,6,3,1] => [1,2,7,3,4,6,5] => ? = 1 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => [2,1,7,3,4,6,5] => ? = 1 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,3,5,7,6,4,1] => [1,2,3,7,4,6,5] => ? = 1 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => [2,1,3,7,4,6,5] => ? = 1 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => [2,3,1,7,4,6,5] => ? = 1 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [2,3,4,7,6,5,1] => [1,2,3,4,7,6,5] => ? = 1 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [1,3,4,7,6,5,2] => [2,1,3,4,7,6,5] => ? = 1 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,2,4,7,6,5,3] => [2,3,1,4,7,6,5] => ? = 1 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [2,3,4,1,7,6,5] => ? = 1 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,6,7,2,5,1,4] => [7,5,2,1,6,3,4] => ? = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [4,6,7,2,5,1,3] => [7,5,1,2,6,3,4] => ? = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [4,6,7,3,5,1,2] => [7,1,5,2,6,3,4] => ? = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [5,6,7,2,4,1,3] => [7,5,1,6,2,3,4] => ? = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [5,6,7,3,4,1,2] => [7,1,5,6,2,3,4] => ? = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,5,7,2,6,1,4] => [7,5,2,1,3,6,4] => ? = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [4,5,7,2,6,1,3] => [7,5,1,2,3,6,4] => ? = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [4,5,7,3,6,1,2] => [7,1,5,2,3,6,4] => ? = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,5,7,4,6,1,3] => [7,2,1,5,3,6,4] => ? = 1 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [3,5,7,4,6,1,2] => [7,1,2,5,3,6,4] => ? = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,4,7,2,6,1,5] => [7,5,2,3,1,6,4] => ? = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,7,3,6,1,5] => [7,2,5,3,1,6,4] => ? = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [1,4,7,3,6,2,5] => [2,7,5,3,1,6,4] => ? = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [2,5,7,3,6,1,4] => [7,2,5,1,3,6,4] => ? = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [1,5,7,3,6,2,4] => [2,7,5,1,3,6,4] => ? = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [1,5,7,4,6,2,3] => [2,7,1,5,3,6,4] => ? = 1 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,6,7,4,5,1,3] => [7,2,1,5,6,3,4] => ? = 1 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [3,6,7,4,5,1,2] => [7,1,2,5,6,3,4] => ? = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,6,7,3,5,1,4] => [7,2,5,1,6,3,4] => ? = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,6,7,3,5,2,4] => [2,7,5,1,6,3,4] => ? = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,7,4,5,2,3] => [2,7,1,5,6,3,4] => ? = 1 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,6,7,3,5,2,1] => [1,7,5,2,6,3,4] => ? = 1 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => ? = 1 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [5,6,7,2,4,3,1] => [1,5,7,6,2,3,4] => ? = 1 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [5,6,7,1,4,3,2] => [5,1,7,6,2,3,4] => ? = 1 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [3,6,7,2,5,4,1] => [1,5,2,7,6,3,4] => ? = 1 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [4,6,7,2,5,3,1] => [1,5,7,2,6,3,4] => ? = 1 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [4,6,7,1,5,3,2] => [5,1,7,2,6,3,4] => ? = 1 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [2,6,7,1,5,4,3] => [5,2,1,7,6,3,4] => ? = 1 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [3,6,7,1,5,4,2] => [5,1,2,7,6,3,4] => ? = 1 + 2
[[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => [2,1,7,6,5,4,3] => 5 = 3 + 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => 6 = 4 + 2
[[1,4,6,7,8],[2],[3],[5]]
=> [5,3,2,1,4,6,7,8] => [3,5,6,7,8,4,2,1] => [1,8,2,7,3,4,5,6] => 3 = 1 + 2
[[1,4,5,6,8],[2],[3],[7]]
=> [7,3,2,1,4,5,6,8] => [3,4,5,7,8,6,2,1] => [1,8,2,3,4,7,5,6] => 3 = 1 + 2
[[1,4,6,8],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6,8] => [3,5,7,8,4,6,2,1] => [1,8,2,6,3,7,4,5] => 3 = 1 + 2
[[1,4,6,8],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6,8] => [3,5,7,8,6,4,2,1] => [1,8,2,7,3,6,4,5] => 4 = 2 + 2
[[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,5,8,1,2,7,6,3] => [5,6,1,2,3,8,7,4] => 3 = 1 + 2
[[1,2,5],[3,6],[4],[7],[8]]
=> [8,7,4,3,6,1,2,5] => [1,4,8,3,7,6,5,2] => [2,1,5,3,8,7,6,4] => 4 = 2 + 2
[[1,2,4],[3,6],[5],[7],[8]]
=> [8,7,5,3,6,1,2,4] => [1,5,8,3,7,6,4,2] => [2,1,5,8,3,7,6,4] => 4 = 2 + 2
[[1,3],[2,4],[5,7],[6,8]]
=> [6,8,5,7,2,4,1,3] => [6,8,5,7,2,4,1,3] => [8,6,1,7,4,2,5,3] => 4 = 2 + 2
[[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [8,1,6,7,4,5,2,3] => 4 = 2 + 2
[[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [3,8,1,7,6,5,4,2] => [4,1,2,8,7,6,5,3] => 5 = 3 + 2
[[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,8,6,5,4,3,2,1] => [1,8,7,6,5,4,2,3] => 6 = 4 + 2
[[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [2,1,8,7,6,5,4,3] => 6 = 4 + 2
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000337
Mp00081: Standard tableaux reading word permutationPermutations
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000337: Permutations ⟶ ℤResult quality: 5% values known / values provided: 5%distinct values known / distinct values provided: 62%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 3 = 1 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,1,2,3,5] => 3 = 1 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [4,1,2,5,3] => 3 = 1 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [3,1,5,2,4] => 3 = 1 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,5,1,3,4] => 3 = 1 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,1,2,3,4] => 4 = 2 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [4,1,2,3,5,6] => 3 = 1 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4,6] => [4,1,2,5,3,6] => 3 = 1 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,4,2,1,3,6] => [3,1,5,2,4,6] => 3 = 1 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [5,4,3,1,2,6] => [2,5,1,3,4,6] => 3 = 1 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [6,3,2,1,4,5] => [4,1,2,5,6,3] => 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,4,2,1,3,5] => [3,1,5,2,6,4] => 3 = 1 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [6,4,3,1,2,5] => [2,5,1,3,6,4] => 3 = 1 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [6,5,2,1,3,4] => [3,1,4,6,2,5] => 3 = 1 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [6,5,3,1,2,4] => [2,4,1,6,3,5] => 3 = 1 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [6,5,4,1,2,3] => [2,3,6,1,4,5] => 3 = 1 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,2,5,1,4] => [4,5,3,6,1,2] => 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,6,2,5,1,4] => [4,5,3,6,1,2] => 3 = 1 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,6,2,5,1,4] => [4,5,3,6,1,2] => 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [3,6,2,5,1,4] => [4,5,3,6,1,2] => 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,6,2,5,1,4] => [4,5,3,6,1,2] => 3 = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,2,6,1,5] => [5,4,2,3,6,1] => 3 = 1 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [4,3,2,6,1,5] => [5,4,2,3,6,1] => 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [4,3,2,6,1,5] => [5,4,2,3,6,1] => 3 = 1 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [4,3,2,6,1,5] => [5,4,2,3,6,1] => 3 = 1 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [6,3,2,5,1,4] => [4,5,2,6,1,3] => 3 = 1 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [6,3,2,5,1,4] => [4,5,2,6,1,3] => 3 = 1 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [6,3,2,5,1,4] => [4,5,2,6,1,3] => 3 = 1 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,5,2,4,1,3] => [3,4,6,1,2,5] => 3 = 1 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [6,5,2,4,1,3] => [3,4,6,1,2,5] => 3 = 1 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,1,2,3,4,6] => 4 = 2 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [6,4,3,2,1,5] => [5,1,2,3,6,4] => 4 = 2 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [6,5,3,2,1,4] => [4,1,2,6,3,5] => 4 = 2 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [6,5,4,2,1,3] => [3,1,6,2,4,5] => 4 = 2 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [6,5,4,3,1,2] => [2,6,1,3,4,5] => 4 = 2 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,1,2,3,4,5] => 5 = 3 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,1,2,3,5,6,7] => 3 = 1 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [5,3,2,1,4,6,7] => [4,1,2,5,3,6,7] => ? = 1 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [5,4,2,1,3,6,7] => [3,1,5,2,4,6,7] => ? = 1 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [5,4,3,1,2,6,7] => [2,5,1,3,4,6,7] => 3 = 1 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [6,3,2,1,4,5,7] => [4,1,2,5,6,3,7] => ? = 1 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => [3,1,5,2,6,4,7] => ? = 1 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [6,4,3,1,2,5,7] => [2,5,1,3,6,4,7] => ? = 1 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [6,5,2,1,3,4,7] => [3,1,4,6,2,5,7] => ? = 1 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [6,5,3,1,2,4,7] => [2,4,1,6,3,5,7] => ? = 1 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [6,5,4,1,2,3,7] => [2,3,6,1,4,5,7] => ? = 1 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [4,1,2,5,6,7,3] => ? = 1 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [3,1,5,2,6,7,4] => ? = 1 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [7,4,3,1,2,5,6] => [2,5,1,3,6,7,4] => ? = 1 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [7,5,2,1,3,4,6] => [3,1,4,6,2,7,5] => ? = 1 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [7,5,3,1,2,4,6] => [2,4,1,6,3,7,5] => ? = 1 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [7,5,4,1,2,3,6] => [2,3,6,1,4,7,5] => ? = 1 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [7,6,2,1,3,4,5] => [3,1,4,5,7,2,6] => ? = 1 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [7,6,3,1,2,4,5] => [2,4,1,5,7,3,6] => ? = 1 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [7,6,4,1,2,3,5] => [2,3,5,1,7,4,6] => ? = 1 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [2,3,4,7,1,5,6] => 3 = 1 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,6,2,5,1,4,7] => [4,5,3,6,1,2,7] => ? = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [3,6,2,5,1,4,7] => [4,5,3,6,1,2,7] => ? = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [3,6,2,5,1,4,7] => [4,5,3,6,1,2,7] => ? = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [3,6,2,5,1,4,7] => [4,5,3,6,1,2,7] => ? = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [3,6,2,5,1,4,7] => [4,5,3,6,1,2,7] => ? = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,7,2,5,1,4,6] => [4,5,3,6,1,7,2] => ? = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [3,7,2,5,1,4,6] => [4,5,3,6,1,7,2] => ? = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [3,7,2,5,1,4,6] => [4,5,3,6,1,7,2] => ? = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [5,7,2,4,1,3,6] => [3,4,6,1,5,7,2] => ? = 1 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [5,7,2,4,1,3,6] => [3,4,6,1,5,7,2] => ? = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,7,2,6,1,4,5] => [4,6,3,5,7,1,2] => ? = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [4,7,2,6,1,3,5] => [3,6,5,4,7,1,2] => ? = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [4,7,3,6,1,2,5] => [2,5,6,4,7,1,3] => ? = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [4,7,2,6,1,3,5] => [3,6,5,4,7,1,2] => ? = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [4,7,3,6,1,2,5] => [2,5,6,4,7,1,3] => ? = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [4,7,3,6,1,2,5] => [2,5,6,4,7,1,3] => ? = 1 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [5,7,2,4,1,3,6] => [3,4,6,1,5,7,2] => ? = 1 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [5,7,2,4,1,3,6] => [3,4,6,1,5,7,2] => ? = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [4,7,2,6,1,3,5] => [3,6,5,4,7,1,2] => ? = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [4,7,3,6,1,2,5] => [2,5,6,4,7,1,3] => ? = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [4,7,3,6,1,2,5] => [2,5,6,4,7,1,3] => ? = 1 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,3,2,6,1,5,7] => [5,4,2,3,6,1,7] => ? = 1 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [4,3,2,6,1,5,7] => [5,4,2,3,6,1,7] => ? = 1 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [4,3,2,6,1,5,7] => [5,4,2,3,6,1,7] => ? = 1 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [4,3,2,6,1,5,7] => [5,4,2,3,6,1,7] => ? = 1 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [6,3,2,5,1,4,7] => [4,5,2,6,1,3,7] => ? = 1 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [6,3,2,5,1,4,7] => [4,5,2,6,1,3,7] => ? = 1 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [6,3,2,5,1,4,7] => [4,5,2,6,1,3,7] => ? = 1 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [6,5,2,4,1,3,7] => [3,4,6,1,2,5,7] => ? = 1 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [6,5,2,4,1,3,7] => [3,4,6,1,2,5,7] => ? = 1 + 2
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [4,3,2,7,1,5,6] => [5,4,2,3,6,7,1] => ? = 1 + 2
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [5,3,2,7,1,4,6] => [4,5,2,6,3,7,1] => ? = 1 + 2
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [5,4,2,7,1,3,6] => [3,5,6,2,4,7,1] => ? = 1 + 2
[[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => [5,1,2,3,4,6,7] => 4 = 2 + 2
[[1,5,7],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7] => [6,4,3,2,1,5,7] => [5,1,2,3,6,4,7] => 4 = 2 + 2
[[1,2,7],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7] => [6,5,4,3,1,2,7] => [2,6,1,3,4,5,7] => 4 = 2 + 2
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [7,4,3,2,1,5,6] => [5,1,2,3,6,7,4] => 4 = 2 + 2
[[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,1,2,3,4,5,7] => 5 = 3 + 2
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [7,5,4,3,2,1,6] => [6,1,2,3,4,7,5] => 5 = 3 + 2
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => [5,1,2,3,7,4,6] => 5 = 3 + 2
[[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [2,7,1,3,4,5,6] => 5 = 3 + 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => 6 = 4 + 2
[[1,3,5,7],[2],[4],[6],[8]]
=> [8,6,4,2,1,3,5,7] => [8,6,4,2,1,3,5,7] => [3,1,5,2,7,4,8,6] => 4 = 2 + 2
[[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [7,1,2,3,4,5,6,8] => 6 = 4 + 2
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000689
Mp00083: Standard tableaux shapeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000689: Dyck paths ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 38%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,6],[3],[5],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,6],[4],[5],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,4,5],[2],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,5],[3],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,5],[4],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[1,5,7],[2,6],[3],[4]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,4,7],[2,6],[3],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,7],[2,6],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,7],[3,6],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,4,7],[2,5],[3],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,7],[2,5],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,7],[3,5],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,7],[2,4],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,7],[3,4],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,5,6],[2,7],[3],[4]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,4,6],[2,7],[3],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,6],[2,7],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,6],[3,7],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,4,5],[2,7],[3],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,5],[2,7],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,5],[3,7],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,4],[2,7],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,4],[3,7],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,3],[4,7],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,4,6],[2,5],[3],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,6],[2,5],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,6],[3,5],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,6],[2,4],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,6],[3,4],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,4,5],[2,6],[3],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,5],[2,6],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,5],[3,6],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,4],[2,6],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,4],[3,6],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,3],[4,6],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,5],[2,4],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,5],[3,4],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3,4],[2,5],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,4],[3,5],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,6],[2,5,9],[4,8],[7]]
=> [3,3,2,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$. This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid. An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Mp00081: Standard tableaux reading word permutationPermutations
Mp00126: Permutations cactus evacuationPermutations
Mp00066: Permutations inversePermutations
St000863: Permutations ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 38%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 1 + 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [5,4,3,1,2] => 4 = 1 + 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [5,4,1,3,2] => 4 = 1 + 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [5,1,4,3,2] => 4 = 1 + 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 1 + 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 5 = 2 + 3
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 4 = 1 + 3
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,5,6,4,2,1] => [6,5,1,4,2,3] => 4 = 1 + 3
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [2,5,6,4,3,1] => [6,1,5,4,2,3] => 4 = 1 + 3
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,5,6,4,3,2] => [1,6,5,4,2,3] => 4 = 1 + 3
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,4,6,5,2,1] => [6,5,1,2,4,3] => 4 = 1 + 3
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,5,3,1] => [6,1,5,2,4,3] => 4 = 1 + 3
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,4,6,5,3,2] => [1,6,5,2,4,3] => 4 = 1 + 3
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [2,3,6,5,4,1] => [6,1,2,5,4,3] => 4 = 1 + 3
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => 4 = 1 + 3
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 4 = 1 + 3
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,2,5,1,4] => [5,3,1,6,4,2] => 4 = 1 + 3
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,5,1,3] => [5,3,6,1,4,2] => 4 = 1 + 3
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [4,6,3,5,1,2] => [5,6,3,1,4,2] => 4 = 1 + 3
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,6,2,4,1,3] => [5,3,6,4,1,2] => 4 = 1 + 3
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 4 = 1 + 3
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,6,3,5,2,1] => [6,5,3,1,4,2] => 4 = 1 + 3
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,6,3,4,2,1] => [6,5,3,4,1,2] => 4 = 1 + 3
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [5,6,2,4,3,1] => [6,3,5,4,1,2] => 4 = 1 + 3
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,6,1,4,3,2] => [3,6,5,4,1,2] => 4 = 1 + 3
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,5,4,1] => [6,3,1,5,4,2] => 4 = 1 + 3
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,6,2,5,3,1] => [6,3,5,1,4,2] => 4 = 1 + 3
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,6,1,5,3,2] => [3,6,5,1,4,2] => 4 = 1 + 3
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [2,6,1,5,4,3] => [3,1,6,5,4,2] => 4 = 1 + 3
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,6,1,5,4,2] => [3,6,1,5,4,2] => 4 = 1 + 3
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 5 = 2 + 3
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,5,3,2,1] => [6,5,4,1,3,2] => 5 = 2 + 3
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,6,5,4,2,1] => [6,5,1,4,3,2] => 5 = 2 + 3
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [2,6,5,4,3,1] => [6,1,5,4,3,2] => 5 = 2 + 3
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 5 = 2 + 3
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 6 = 3 + 3
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 1 + 3
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,6,7,4,2,1] => [7,6,1,5,2,3,4] => ? = 1 + 3
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,6,7,4,3,1] => [7,1,6,5,2,3,4] => ? = 1 + 3
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,5,6,7,4,3,2] => [1,7,6,5,2,3,4] => ? = 1 + 3
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [3,4,6,7,5,2,1] => [7,6,1,2,5,3,4] => ? = 1 + 3
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => [7,1,6,2,5,3,4] => ? = 1 + 3
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,4,6,7,5,3,2] => [1,7,6,2,5,3,4] => ? = 1 + 3
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [2,3,6,7,5,4,1] => [7,1,2,6,5,3,4] => ? = 1 + 3
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => ? = 1 + 3
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,2,6,7,5,4,3] => [1,2,7,6,5,3,4] => 4 = 1 + 3
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [3,4,5,7,6,2,1] => [7,6,1,2,3,5,4] => ? = 1 + 3
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,5,7,6,3,1] => [7,1,6,2,3,5,4] => ? = 1 + 3
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => [1,7,6,2,3,5,4] => ? = 1 + 3
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 1 + 3
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => [1,7,2,6,3,5,4] => ? = 1 + 3
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => [1,2,7,6,3,5,4] => 4 = 1 + 3
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 1 + 3
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [1,3,4,7,6,5,2] => [1,7,2,3,6,5,4] => ? = 1 + 3
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => 4 = 1 + 3
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => 4 = 1 + 3
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,6,7,2,5,1,4] => [6,4,1,7,5,2,3] => ? = 1 + 3
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [4,6,7,2,5,1,3] => [6,4,7,1,5,2,3] => ? = 1 + 3
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [4,6,7,3,5,1,2] => [6,7,4,1,5,2,3] => ? = 1 + 3
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [5,6,7,2,4,1,3] => [6,4,7,5,1,2,3] => ? = 1 + 3
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [5,6,7,3,4,1,2] => [6,7,4,5,1,2,3] => ? = 1 + 3
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,5,7,2,6,1,4] => [6,4,1,7,2,5,3] => ? = 1 + 3
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [4,5,7,2,6,1,3] => [6,4,7,1,2,5,3] => ? = 1 + 3
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [4,5,7,3,6,1,2] => [6,7,4,1,2,5,3] => ? = 1 + 3
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,5,7,4,6,1,3] => [6,1,7,4,2,5,3] => ? = 1 + 3
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [3,5,7,4,6,1,2] => [6,7,1,4,2,5,3] => ? = 1 + 3
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,4,7,2,6,1,5] => [6,4,1,2,7,5,3] => ? = 1 + 3
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,7,3,6,1,5] => [6,1,4,2,7,5,3] => ? = 1 + 3
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [1,4,7,3,6,2,5] => [1,6,4,2,7,5,3] => ? = 1 + 3
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [2,5,7,3,6,1,4] => [6,1,4,7,2,5,3] => ? = 1 + 3
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [1,5,7,3,6,2,4] => [1,6,4,7,2,5,3] => ? = 1 + 3
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [1,5,7,4,6,2,3] => [1,6,7,4,2,5,3] => ? = 1 + 3
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,6,7,4,5,1,3] => [6,1,7,4,5,2,3] => ? = 1 + 3
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [3,6,7,4,5,1,2] => [6,7,1,4,5,2,3] => ? = 1 + 3
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,6,7,3,5,1,4] => [6,1,4,7,5,2,3] => ? = 1 + 3
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,6,7,3,5,2,4] => [1,6,4,7,5,2,3] => ? = 1 + 3
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,7,4,5,2,3] => [1,6,7,4,5,2,3] => ? = 1 + 3
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,6,7,3,5,2,1] => [7,6,4,1,5,2,3] => ? = 1 + 3
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [5,6,7,3,4,2,1] => [7,6,4,5,1,2,3] => ? = 1 + 3
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [5,6,7,2,4,3,1] => [7,4,6,5,1,2,3] => ? = 1 + 3
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [5,6,7,1,4,3,2] => [4,7,6,5,1,2,3] => ? = 1 + 3
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [3,6,7,2,5,4,1] => [7,4,1,6,5,2,3] => ? = 1 + 3
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [4,6,7,2,5,3,1] => [7,4,6,1,5,2,3] => ? = 1 + 3
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [4,6,7,1,5,3,2] => [4,7,6,1,5,2,3] => ? = 1 + 3
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [2,6,7,1,5,4,3] => [4,1,7,6,5,2,3] => ? = 1 + 3
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [3,6,7,1,5,4,2] => [4,7,1,6,5,2,3] => ? = 1 + 3
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [4,5,7,3,6,2,1] => [7,6,4,1,2,5,3] => ? = 1 + 3
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,7,4,6,2,1] => [7,6,1,4,2,5,3] => ? = 1 + 3
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,7,4,6,3,1] => [7,1,6,4,2,5,3] => ? = 1 + 3
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,7,4,6,3,2] => [1,7,6,4,2,5,3] => ? = 1 + 3
[[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => [1,3,7,2,6,5,4] => [1,4,2,7,6,5,3] => 4 = 1 + 3
[[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => 5 = 2 + 3
Description
The length of the first row of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the length of the first row of $P$ and $Q$.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St001566The length of the longest arithmetic progression in a permutation. St001644The dimension of a graph. St000365The number of double ascents of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000702The number of weak deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St000051The size of the left subtree of a binary tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St000080The rank of the poset. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000094The depth of an ordered tree. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001963The tree-depth of a graph.