Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001604
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: 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]
 => 0
[[1,2,3,4,5]]
 => [5]
 => [2,2,1]
 => [2,1]
 => 0
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,3,4,5,6]]
 => [6]
 => [2,2,2]
 => [2,2]
 => 1
[[1,3,4,5,6],[2]]
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,4,5,6],[3]]
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,3,5,6],[4]]
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,3,4,6],[5]]
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,3,4,5],[6]]
 => [5,1]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,3,5],[2,4,6]]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[[1,2,5],[3,4,6]]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[[1,3,4],[2,5,6]]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[[1,2,4],[3,5,6]]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[[1,2,3],[4,5,6]]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [3,3]
 => [3]
 => 1
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001171
(load all 2 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ? = 0 + 1
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => ? = 0 + 1
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ? = 0 + 1
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 1 + 1
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => ? = 0 + 1
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => ? = 0 + 1
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => ? = 0 + 1
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => ? = 0 + 1
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => ? = 0 + 1
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ? = 0 + 1
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ? = 0 + 1
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ? = 0 + 1
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ? = 0 + 1
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ? = 0 + 1
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ? = 0 + 1
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => ? = 0 + 1
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => ? = 0 + 1
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => ? = 0 + 1
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => ? = 0 + 1
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => ? = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 1 + 1
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => ? = 1 + 1
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 1 + 1
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 1 + 1
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 1 + 1
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 1 + 1
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 1 + 1
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 1 + 1
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => ? = 0 + 1
Description
The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001000
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 2
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 2
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 2
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 2
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 2
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 2
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 2
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 2
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 2
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 2
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 2
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 + 2
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 2
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 2
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001424
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001424: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 2
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 0 + 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 0 + 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 0 + 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 0 + 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 0 + 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 0 + 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 0 + 2
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 0 + 2
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 11111100000010 => ? = 1 + 2
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 2
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 2
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 2
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 2
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 2
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 2
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 2
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 2
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 2
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 3 = 1 + 2
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 2
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 2
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 2
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 2
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 2
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 2
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 1 + 2
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 1111111000000010 => ? = 1 + 2
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 2
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 2
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 2
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 2
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 2
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 2
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 2
Description
The number of distinct squares in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uu$ is a factor of the word. Note that every word of length at least four contains a square.
Matching statistic: St001712
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 0 + 2
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 0 + 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 0 + 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 0 + 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 0 + 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 0 + 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 0 + 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 0 + 2
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 0 + 2
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 2
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 2
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 2
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 2
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 2
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 2
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 1 + 2
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 2
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 2
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 2
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 2
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 2
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 2
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => ? = 1 + 2
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
 => ? = 1 + 2
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 2
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 2
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001200
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 3
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 + 3
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001355
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 3
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 0 + 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3 = 0 + 3
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3 = 0 + 3
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3 = 0 + 3
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3 = 0 + 3
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3 = 0 + 3
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3 = 0 + 3
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 0 + 3
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 11111100000010 => ? = 1 + 3
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 3
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 3
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 3
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 3
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 3
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 3
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 3
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 3
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 3
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 3
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 1 + 3
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 3
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 3
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 3
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 3
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 3
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 3
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 1 + 3
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 1111111000000010 => ? = 1 + 3
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 3
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 3
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 3
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 3
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 3
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 3
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 3
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Matching statistic: St001462
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 0 + 3
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 0 + 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 3 = 0 + 3
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 3 = 0 + 3
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 3 = 0 + 3
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 3 = 0 + 3
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 3 = 0 + 3
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 3 = 0 + 3
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => ? = 0 + 3
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 3
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,11],[5,7,8,9,10,12]]
 => ? = 0 + 3
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 3
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 3
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 3
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 3
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 0 + 3
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 1 + 3
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 3
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 3
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 3
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 3
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 3
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 0 + 3
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
 => ? = 1 + 3
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
 => ? = 1 + 3
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
 => ? = 1 + 3
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,11],[5,6,8,9,10,12]]
 => ? = 0 + 3
Description
The number of factors of a standard tableaux under concatenation. The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10]. This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St001553
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 0 + 3
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 3
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 0 + 3
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 1 + 3
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 3
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 3
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 + 3
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 3
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 0 + 3
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St001267
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 4
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 0 + 4
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 0 + 4
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 0 + 4
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 0 + 4
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 0 + 4
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 0 + 4
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 0 + 4
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 0 + 4
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 11111100000010 => ? = 1 + 4
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 4
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 4
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 4
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 4
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => ? = 0 + 4
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 4
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 4
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 4
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 4
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 0 + 4
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 5 = 1 + 4
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 + 4
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 4
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 4
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 4
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 4
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0 + 4
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 1 + 4
[[1,2,3,4,5,6,7]]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 1111111000000010 => ? = 1 + 4
[[1,3,4,5,6,7],[2]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 4
[[1,2,4,5,6,7],[3]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 4
[[1,2,3,5,6,7],[4]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 4
[[1,2,3,4,6,7],[5]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 4
[[1,2,3,4,5,7],[6]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 4
[[1,2,3,4,5,6],[7]]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 11111010000010 => ? = 1 + 4
[[1,3,5,6,7],[2,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,5,6,7],[3,4]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,3,4,6,7],[2,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,4,6,7],[3,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,3,6,7],[4,5]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,3,4,5,7],[2,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,4,5,7],[3,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,3,5,7],[4,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,3,4,7],[5,6]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,3,4,5,6],[2,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,4,5,6],[3,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,3,5,6],[4,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
[[1,2,3,4,6],[5,7]]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => ? = 0 + 4
Description
The length of the Lyndon factorization of the binary word. The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001417The length of a longest palindromic subword of a binary word. St000391The sum of the positions of the ones in a binary word.