searching the database
Your data matches 53 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001621
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001878
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001232
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{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 = 2 + 1
{{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 + 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{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 = 2 + 1
{{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 + 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{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]
=> ? = 2 + 1
{{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 + 1
{{1,2},{3},{4,6},{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]
=> ? = 2 + 1
{{1,3,4},{2,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3,5},{2,4,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3,6},{2,4,5}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{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 = 2 + 1
{{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 + 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
{{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 = 2 + 1
{{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 + 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{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]
=> ? = 2 + 1
{{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 + 1
{{1,3},{2},{4,6},{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]
=> ? = 2 + 1
{{1,4,5},{2,3,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{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 = 2 + 1
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{1,2,3},{4,7},{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{1,2,4},{3,7},{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{1,2},{3,4,7},{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{1,2,5},{3,7},{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 = 2 + 1
{{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 = 2 + 1
{{1,2},{3,5,6},{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 = 2 + 1
{{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 = 2 + 1
{{1,2},{3,5,7},{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 = 2 + 1
{{1,2},{3,5},{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 = 2 + 1
{{1,2,6},{3,7},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
{{1,2,7},{3,6},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,6,7},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,6},{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 = 2 + 1
{{1,2},{3,7},{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 = 2 + 1
{{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 = 2 + 1
{{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 = 2 + 1
{{1,3,4},{2,7},{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 = 2 + 1
{{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 = 2 + 1
{{1,3},{2,4,5},{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 = 2 + 1
{{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 = 2 + 1
{{1,3},{2,4,6},{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 = 2 + 1
{{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 = 2 + 1
{{1,3},{2,4,7},{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 = 2 + 1
{{1,3},{2,4},{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 = 2 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 100%
Values
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,2,3},{4,5,6}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,4},{3,5,6}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,5},{3,4,6}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,6},{3,4,5}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,3,4},{2,5,6}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,5},{2,4,6}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,6},{2,4,5}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,4,5},{2,3,6}}
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,4,5},{2,3},{6}}
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
{{1,4},{2,3,5},{6}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,4},{2,3,6},{5}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,5},{2,3,4},{6}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,6},{2,3,4},{5}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,5},{2,3,6},{4}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,6},{2,3,5},{4}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,4},{2,5,6},{3}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,5},{2,4,6},{3}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,6},{2,4,5},{3}}
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,3},{4,5,6,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,4},{3,5,6,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,4},{3,5,6},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,4},{3,5,7},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,4},{3,6,7},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,5},{3,4,6,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,4,6},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,5},{3,4,7},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,6},{3,4,5,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,6},{3,4,5},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,7},{3,4,5,6}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,7},{3,4,5},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,6},{3,4,7},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,7},{3,4,6},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,5},{3,6,7},{4}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,6},{3,5,7},{4}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,2,7},{3,5,6},{4}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,4},{2,5,6,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,3,4},{2,5,6},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,4},{2,5,7},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,4},{2,6,7},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,5},{2,4,6,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,3,5},{2,4,6},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,5},{2,4,7},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,6},{2,4,5,7}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,3,6},{2,4,5},{7}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,7},{2,4,5,6}}
=> [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,3,7},{2,4,5},{6}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,6},{2,4,7},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,7},{2,4,6},{5}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,5},{2,6,7},{4}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
{{1,3,6},{2,5,7},{4}}
=> [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
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: St001195
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,3,4},{2,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3,5},{2,4,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3,6},{2,4,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,4,5},{2,3,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,4,6},{2,3,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
{{1,2,3,4},{5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,6},{4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,7},{4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4,5},{3,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4,6},{3,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4,7},{3,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,5,6},{3,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,5,7},{3,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,5},{3,4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,6,7},{3,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,6},{3,4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,7},{3,4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4,5},{2,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4,6},{2,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4,7},{2,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,5,6},{2,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,5,7},{2,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,5},{2,4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,6,7},{2,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,6},{2,4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,7},{2,4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,5,6},{2,3,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,5,7},{2,3,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,5},{2,3,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,6,7},{2,3,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,6},{2,3,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,7},{2,3,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,5,6,7},{2,3,4}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,5,6},{2,3,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,5,7},{2,3,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,6,7},{2,3,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1},{2},{3,4,5},{6},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1},{2},{3,4,6},{5},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1},{2},{3,4,7},{5},{6}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1},{2},{3,5,6},{4},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001001
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3,4},{2,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3,5},{2,4,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3,6},{2,4,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,4,5},{2,3,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,4,6},{2,3,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
{{1,2,3,4},{5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,3,5},{4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,3,6},{4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,3,7},{4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,3},{4,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,4,5},{3,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,4,6},{3,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,4,7},{3,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,4},{3,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,5,6},{3,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,5,7},{3,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,5},{3,4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,6,7},{3,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,6},{3,4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,7},{3,4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,4,5},{2,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,4,6},{2,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,4,7},{2,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,4},{2,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,5,6},{2,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,5,7},{2,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,5},{2,4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,6,7},{2,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,6},{2,4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,3,7},{2,4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,4,5,6},{2,3,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,4,5,7},{2,3,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,4,5},{2,3,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,4,6,7},{2,3,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,4,6},{2,3,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,4,7},{2,3,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,5,6,7},{2,3,4}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,5,6},{2,3,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,5,7},{2,3,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,6,7},{2,3,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4,5},{6},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4,6},{5},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4,7},{5},{6}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
{{1},{2},{3,5,6},{4},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001515
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
{{1,2,3},{4,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,4},{3,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,5},{3,4,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,6},{3,4,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3,4},{2,5,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3,5},{2,4,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3,6},{2,4,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,4,5},{2,3,6}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,4,6},{2,3,5}}
=> [3,3]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 3
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
{{1,2,3,4},{5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,3,5},{4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,3,6},{4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,3,7},{4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,3},{4,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,4,5},{3,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,4,6},{3,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,4,7},{3,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,4},{3,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,5,6},{3,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,5,7},{3,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,5},{3,4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,6,7},{3,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,6},{3,4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,2,7},{3,4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,4,5},{2,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,4,6},{2,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,4,7},{2,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,4},{2,5,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,5,6},{2,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,5,7},{2,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,5},{2,4,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,6,7},{2,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,6},{2,4,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,3,7},{2,4,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,4,5,6},{2,3,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,4,5,7},{2,3,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,4,5},{2,3,6,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,4,6,7},{2,3,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,4,6},{2,3,5,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,4,7},{2,3,5,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,5,6,7},{2,3,4}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,5,6},{2,3,4,7}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,5,7},{2,3,4,6}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1,6,7},{2,3,4,5}}
=> [4,3]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 1 + 3
{{1},{2},{3,4,5},{6},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 1 + 3
{{1},{2},{3,4,6},{5},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 1 + 3
{{1},{2},{3,4,7},{5},{6}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 1 + 3
{{1},{2},{3,5,6},{4},{7}}
=> [3,1,1,1,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 1 + 3
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000455
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3,4},{2,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3,5},{2,4,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3,6},{2,4,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,4},{5,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,4,5},{2,3,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,4,6},{2,3,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,4},{2,3,5},{6}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,5,6},{2,3,4}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3,4},{5,6,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,5},{4,6,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,6},{4,5,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,7},{4,5,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3},{4,5,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4,5},{3,6,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4,6},{3,5,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4,7},{3,5,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4},{3,5,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4},{3},{5,6,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,5,6},{3,4,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,5,7},{3,4,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,5},{3,4,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,6,7},{3,4,5}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,6},{3,4,5,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,7},{3,4,5,6}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,5},{3},{4,6,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,6},{3},{4,5,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,7},{3},{4,5,6}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4,5},{2,6,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4,6},{2,5,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4,7},{2,5,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4},{2,5,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4},{2},{5,6,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,5,6},{2,4,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,5,7},{2,4,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,5},{2,4,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,6,7},{2,4,5}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,6},{2,4,5,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,7},{2,4,5,6}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,5},{2},{4,6,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,6},{2},{4,5,7}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,7},{2},{4,5,6}}
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,4,5,6},{2,3,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,4,5,7},{2,3,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,4,5},{2,3,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,4,6,7},{2,3,5}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,4,6},{2,3,5,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,4,7},{2,3,5,6}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001645
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,5},{2,4},{3}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1},{2,5},{3,4}}
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 6
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,4},{3,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,5},{3,4,6}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,6},{3,4,5}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,3,4},{2,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3,5},{2,4,6}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3,6},{2,4,5}}
=> [3,3] => [1,1,2,1,1] => ([(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)
=> ? = 1 + 6
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 6
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,5},{3,6,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,5},{3,7},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,6},{3,5,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,6},{3,5},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,7},{3,5,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,7},{3,5},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,6},{3,7},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,5,6},{3,4,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,5,6},{3,4},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,5,7},{3,4,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,5,7},{3,4},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,6,7},{3,4,5}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,6,7},{3,4},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,5,6},{3,7},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,5,7},{3,6},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,2,6,7},{3,5},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,5},{2,6,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,5},{2,6},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,5},{2,7},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,6},{2,5,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,6},{2,5},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,7},{2,5,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,7},{2,5},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,6},{2,7},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,4,7},{2,6},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,5,6},{2,4,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,5,6},{2,4},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,5,7},{2,4,6}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,5,7},{2,4},{6}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,6,7},{2,4,5}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,6,7},{2,4},{5}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,5,6},{2,7},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,5,7},{2,6},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,3,6,7},{2,5},{4}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,4,5,6},{2,3,7}}
=> [4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
{{1,4,5,6},{2,3},{7}}
=> [4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St001964
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,4,5},{2,3,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,4,6},{2,3,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,4},{2,3,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,4},{2,3,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,5,6},{2,3,4}}
=> [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
{{1,5},{2,3,4},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,6},{2,3,4},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1},{2,3,4},{5,6}}
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
{{1,5,6},{2,3},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,5},{2,3,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,6},{2,3,5},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1},{2,3,5},{4,6}}
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
{{1,6},{2,3},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1},{2,3,6},{4,5}}
=> [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
{{1},{2,3},{4,5,6}}
=> [1,2,3] => [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ? = 1 - 1
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 1 - 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
{{1,4,5},{2,6},{3}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,4,6},{2,5},{3}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,4},{2,5,6},{3}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,4},{2,5},{3,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,4},{2,6},{3,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,4},{2},{3,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,4},{2},{3,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,5,6},{2,4},{3}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,5},{2,4,6},{3}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,5},{2,4},{3,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,6},{2,4,5},{3}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
{{1,6},{2,4},{3},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,5},{2,6},{3},{4}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
{{1,6},{2,5},{3},{4}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001413Half the length of the longest even length palindromic prefix of a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St000393The number of strictly increasing runs in a binary word. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000237The number of small exceedances. St000352The Elizalde-Pak rank of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000022The number of fixed points of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000402Half the size of the symmetry class of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000862The number of parts of the shifted shape of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. 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 $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!