Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001604
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => [3]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => [3]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => [3]
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => [2,1]
 => 0
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4]]
 => [4]
 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => [4]
 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => [3]
 => 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => [3,1]
 => 0
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => [3]
 => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[4,4,4],[3,1]]
 => [3,1]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 0
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5,5],[4]]
 => [4]
 => 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4,4],[3]]
 => [3]
 => 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3,1]]
 => [3,1]
 => 0
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,3],[3]]
 => [3]
 => 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [[3,3,3,3,3],[2,1]]
 => [2,1]
 => 0
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [[5,5,5],[4]]
 => [4]
 => 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[3,3,3,3],[2,2]]
 => [2,2]
 => 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => [2,1]
 => 0
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1,1]]
 => [2,1,1]
 => 0
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1,1]]
 => [1,1,1]
 => 0
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [[6,6,6,6,6,6],[5]]
 => [5]
 => 1
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [[6,6,6,6,6],[5]]
 => [5]
 => 1
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,4],[4]]
 => [4]
 => 1
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4,1]]
 => [4,1]
 => 0
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [[5,5,5,4],[4]]
 => [4]
 => 1
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4,1]]
 => [4,1]
 => 0
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [[4,4,3,3],[3]]
 => [3]
 => 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [[4,4,4,4],[3,2]]
 => [3,2]
 => 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3,1]]
 => [3,1]
 => 0
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4,2],[3]]
 => [3]
 => 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1,1]]
 => [3,1,1]
 => 0
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [[4,4,2],[3]]
 => [3]
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => [3,1]
 => 0
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [[4,4,4],[3,2]]
 => [3,2]
 => 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => [2,1]
 => 0
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [[4,4,3],[2,2]]
 => [2,2]
 => 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => [2,1]
 => 0
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [1,1,1]
 => 0
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [[3,3,3,3,3],[2,2]]
 => [2,2]
 => 1
[6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [[3,3,3,3,3],[2,1,1]]
 => [2,1,1]
 => 0
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [[5,5,5],[4,1]]
 => [4,1]
 => 0
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2]]
 => [2,2]
 => 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [[3,3,3,3],[2,2,1]]
 => [2,2,1]
 => 1
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => [2,1]
 => 0
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[5,5],[4]]
 => [4]
 => 1
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: St000353
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 15%
Values
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,1] => [6,2,3,4,1,5] => 2 = 1 + 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => [6,2,3,1,4,5] => 2 = 1 + 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => [6,2,3,5,4,1] => 1 = 0 + 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => ? = 1 + 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,6] => [6,2,1,3,4,5] => 1 = 0 + 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,4,5] => [1,6,2,5,3,4] => 2 = 1 + 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,1,7] => ? = 1 + 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,1,8] => [8,2,3,4,5,1,6,7] => ? = 1 + 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,1,6,7] => [7,2,3,1,4,5,6] => ? = 1 + 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,8,1,7] => [8,2,3,4,5,7,6,1] => ? = 0 + 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => [6,2,5,3,4,1] => 2 = 1 + 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,1,6] => [8,2,3,4,6,7,5,1] => ? = 0 + 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => [1,6,4,5,2,3] => 2 = 1 + 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => [1,6,2,3,4,5] => 1 = 0 + 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,1] => [9,2,3,4,5,6,7,1,8] => ? = 1 + 1
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,8,1,9] => [9,2,3,4,5,6,1,7,8] => ? = 1 + 1
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,6,1,7,8] => [8,2,3,4,1,5,6,7] => ? = 0 + 1
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,7,9,1,8] => [9,2,3,4,5,6,8,7,1] => ? = 1 + 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,7,1,5,6] => [7,2,3,6,4,5,1] => ? = 0 + 1
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,4,5,1,8,6,7] => [8,2,3,1,4,7,5,6] => ? = 1 + 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => [6,4,5,2,3,1] => 2 = 1 + 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,1,5,6,7] => [7,2,1,3,4,5,6] => ? = 0 + 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,4,1,7,8,5,6] => [8,2,1,3,6,7,4,5] => ? = 0 + 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => [6,4,2,3,5,1] => 1 = 0 + 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => [6,5,2,3,4,1] => 1 = 0 + 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,10,1] => [10,2,3,4,5,6,7,8,1,9] => ? = 1 + 1
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,8,9,1,10] => [10,2,3,4,5,6,7,1,8,9] => ? = 1 + 1
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,6,7,1,8,9] => [9,2,3,4,5,1,6,7,8] => ? = 1 + 1
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,7,8,10,1,9] => [10,2,3,4,5,6,7,9,8,1] => ? = 0 + 1
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,5,8,1,6,7] => [8,2,3,4,7,5,6,1] => ? = 1 + 1
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,9,10,1,8] => [10,2,3,4,5,6,8,9,7,1] => ? = 0 + 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,6,7,1,4,5] => [7,2,5,6,3,4,1] => ? = 1 + 1
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,4,7,1,8,5,6] => [8,2,3,1,6,7,4,5] => ? = 1 + 1
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1,6,7,8] => [8,2,3,1,4,5,6,7] => ? = 0 + 1
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,4,5,1,8,9,6,7] => [9,2,3,1,4,7,8,5,6] => ? = 1 + 1
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,8,9,10,1,7] => [10,2,3,4,5,7,8,9,6,1] => ? = 0 + 1
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,6,1,4,5,7] => [7,2,5,3,4,6,1] => ? = 1 + 1
[5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,6,1,7,8,4,5] => [8,2,1,5,6,7,3,4] => ? = 0 + 1
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,4,1,5,8,6,7] => [8,2,1,3,4,7,5,6] => ? = 1 + 1
[4,4,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [4,5,1,6,7,2,3] => [3,1,4,5,6,7,2] => ? = 0 + 1
[3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,2,3,6,4,5] => [6,2,5,1,3,4] => 2 = 1 + 1
[3,3,2,2]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,6,1,2,3,5] => [3,5,6,2,4,1] => 1 = 0 + 1
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,6,1,2,3,4] => [4,5,6,2,3,1] => 1 = 0 + 1
[6,5]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [2,5,6,7,1,3,4] => [7,4,5,6,2,3,1] => ? = 1 + 1
[6,3,2]
 => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6,8] => ? => ? = 0 + 1
[6,2,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? => ? => ? = 0 + 1
[5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,1,3,4,7] => [7,4,5,2,3,6,1] => ? = 1 + 1
[5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [2,5,6,1,7,8,3,4] => [8,4,1,5,6,7,2,3] => ? = 1 + 1
[5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => [1,7,2,3,4,5,6] => ? = 0 + 1
[5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [2,3,6,1,4,8,5,7] => ? => ? = 1 + 1
[5,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,8,1,5,6,7] => [8,2,3,7,4,5,6,1] => ? = 0 + 1
[5,2,2,1,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,4,1,5,8,9,6,7] => ? => ? = 0 + 1
[4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [4,5,1,2,7,3,6] => [3,4,7,6,1,2,5] => ? = 1 + 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,2,3,4,5] => [5,6,1,2,3,4] => 1 = 0 + 1
[6,5,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,5,6,7,1,8,3,4] => ? => ? = 0 + 1
[6,3,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? => ? => ? = 1 + 1
[5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,5,1,3,4,6,7] => [7,4,2,3,5,6,1] => ? = 0 + 1
[5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [2,5,6,1,3,8,4,7] => ? => ? = 0 + 1
[5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,1,4,5,8,6,7] => ? => ? = 1 + 1
[5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,3,6,8,1,4,5,7] => ? => ? = 0 + 1
[5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [2,3,6,1,4,8,9,5,7] => ? => ? = 1 + 1
[4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [4,1,2,3,7,5,6] => [3,7,2,6,1,4,5] => ? = 1 + 1
[4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,5,7,1,2,3,6] => [3,4,6,7,2,5,1] => ? = 0 + 1
[7,4,1,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? => ? => ? = 1 + 1
[6,5,2]
 => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4,8] => ? => ? = 1 + 1
[4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [4,1,7,2,3,5,6] => [1,3,6,7,2,4,5] => 1 = 0 + 1
Description
The number of inner valleys of a permutation. The number of valleys including the boundary is [[St000099]].