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Matching statistic: St001314
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
Matching statistic: St000422
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 52%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 0 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 0 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 2 = 0 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2 = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 0 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 2 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 2 = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 0 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 4 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 0 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 5 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 2 = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 2 = 0 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 2 = 0 + 2
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000455
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 48%●distinct values known / distinct values provided: 17%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 48%●distinct values known / distinct values provided: 17%
Values
[1]
=> 10 => [1,2] => ([(1,2)],3)
=> 0
[2]
=> 100 => [1,3] => ([(2,3)],4)
=> 0
[1,1]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[3]
=> 1000 => [1,4] => ([(3,4)],5)
=> 0
[2,1]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,1]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[4]
=> 10000 => [1,5] => ([(4,5)],6)
=> 0
[3,1]
=> 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,2]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1]
=> 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[5]
=> 100000 => [1,6] => ([(5,6)],7)
=> 0
[4,1]
=> 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,2]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,1,1]
=> 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,2,1]
=> 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[4,2]
=> 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,3]
=> 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[3,2,1]
=> 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[2,2,2]
=> 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[4,3]
=> 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,3,1]
=> 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,2,2]
=> 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[4,4]
=> 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[3,3,2]
=> 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[3,3,3]
=> 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
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: St000068
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Values
[1]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3]
=> 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[2,1]
=> 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[1,1,1]
=> 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[4]
=> 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[3,1]
=> 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
[2,2]
=> 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[2,1,1]
=> 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 + 1
[1,1,1,1]
=> 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
[5]
=> 100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 0 + 1
[4,1]
=> 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 1 + 1
[3,2]
=> 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
[3,1,1]
=> 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 2 + 1
[2,2,1]
=> 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 2 + 1
[2,1,1,1]
=> 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 3 + 1
[1,1,1,1,1]
=> 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 0 + 1
[4,2]
=> 100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 1 + 1
[3,3]
=> 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
[3,2,1]
=> 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 4 + 1
[2,2,2]
=> 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
[2,2,1,1]
=> 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 5 + 1
[4,3]
=> 101000 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 1 + 1
[3,3,1]
=> 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 1 + 1
[3,2,2]
=> 101100 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 2 + 1
[2,2,2,1]
=> 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 3 + 1
[4,4]
=> 110000 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 0 + 1
[3,3,2]
=> 110100 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> ? = 2 + 1
[2,2,2,2]
=> 111100 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 0 + 1
[3,3,3]
=> 111000 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000882
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000882: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000882: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 0 + 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 3 = 0 + 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 3 = 0 + 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 3 = 0 + 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 4 = 1 + 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 3 = 0 + 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? = 0 + 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ? = 1 + 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 3 = 0 + 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ? = 2 + 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 0 + 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,4,5,8,7,1,6] => ? = 0 + 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? = 1 + 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 4 = 1 + 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ? = 2 + 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ? = 2 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 3 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => ? = 0 + 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 1 + 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ? = 0 + 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 4 + 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? = 0 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 5 + 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? = 1 + 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 1 + 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 2 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 3 + 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,3,4,7,6,1,8,5] => ? = 0 + 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? = 2 + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,5,4,1,6,7,8,3] => ? = 0 + 3
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [2,3,6,5,1,7,8,4] => ? = 0 + 3
Description
The number of connected components of short braid edges in the graph of braid moves of a permutation.
Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for
$$(1243) = (12)(34)(23) = (34)(12)(23)$$
share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$.
This statistic counts the number connected components of such short braid moves among all reduced words.
Matching statistic: St000090
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [[1],[2]]
=> [1,1] => 0
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [1,2,1] => 0
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [2,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [1,2,2,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [2,2,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [1,2,2,2,1] => ? = 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [1,2,3,2] => ? = 1
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [1,3,2,2] => ? = 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [2,2,2,2] => ? = 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [1,2,2,2,2,1] => ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [1,2,2,3,2] => ? = 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [1,4,3] => ? = 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [1,2,3,2,2] => ? = 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [3,3,2] => ? = 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [1,3,2,2,2] => ? = 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [2,2,2,2,2] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [1,2,4,3] => ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [3,2,3] => ? = 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [1,4,3,2] => ? = 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [3,3,2,2] => ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [1,4,2,3] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [3,2,3,2] => ? = 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [1,5,4] => ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [4,4,2] => ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [3,2,2,3] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [3,3,4] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [4,2,4] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 0
Description
The variation of a composition.
Matching statistic: St000091
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [[1],[2]]
=> [2] => 0
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,2] => 0
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [3,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,2,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,2,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,2,2,2] => ? = 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,2,3,1] => ? = 1
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,3,2,1] => ? = 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,2,2,1] => ? = 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,2,2,3,1] => ? = 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,4,2] => ? = 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,2,3,2,1] => ? = 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,3,1] => ? = 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,3,2,2,1] => ? = 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,2,2,2,1] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,2,4,2] => ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,2,2] => ? = 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,4,3,1] => ? = 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,3] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [4,3,2,1] => ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,4,2,2] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [4,2,3,1] => ? = 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,5,3] => ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [5,4,1] => ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [4,2,2,2] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [4,3,3] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [5,2,3] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,4] => ? = 0
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St000233
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 0
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> {{1,2,4},{3,5,6}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 1
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> {{1,3,5,7,9},{2,4,6,8,10}}
=> ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> {{1,3,5,7,8},{2,4,6,9,10}}
=> ? = 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> ? = 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> {{1,3,5,6,8},{2,4,7,9,10}}
=> ? = 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> {{1,3,4,6,8},{2,5,7,9,10}}
=> ? = 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> {{1,2,4,6,8},{3,5,7,9,10}}
=> ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> {{1,3,5,6,7},{2,4,8,9,10}}
=> ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> {{1,2,3,5},{4,6,7,8}}
=> ? = 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> {{1,3,4,5,8},{2,6,7,9,10}}
=> ? = 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> {{1,2,3,6,8},{4,5,7,9,10}}
=> ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> {{1,3,4,5,7},{2,6,8,9,10}}
=> ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> {{1,2,3,5,8},{4,6,7,9,10}}
=> ? = 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> {{1,3,4,5,6},{2,7,8,9,10}}
=> ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> {{1,2,3,4,8},{5,6,7,9,10}}
=> ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> {{1,2,3,5,7},{4,6,8,9,10}}
=> ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> {{1,2,3,5,6},{4,7,8,9,10}}
=> ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> {{1,2,3,4,6},{5,7,8,9,10}}
=> ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> {{1,2,3,4,5},{6,7,8,9,10}}
=> ? = 0
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000650
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [[1],[2]]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? = 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? = 1
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => ? = 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => ? = 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => ? = 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ? = 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,4,7,9,10,1,3,5,6,8] => ? = 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => ? = 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => ? = 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,4,8,9,10,1,3,5,6,7] => ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => ? = 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => ? = 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [4,5,7,9,10,1,2,3,6,8] => ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [2,6,8,9,10,1,3,4,5,7] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [4,6,7,9,10,1,2,3,5,8] => ? = 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [2,7,8,9,10,1,3,4,5,6] => ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> [5,6,7,9,10,1,2,3,4,8] => ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [4,6,8,9,10,1,2,3,5,7] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> [4,7,8,9,10,1,2,3,5,6] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [5,7,8,9,10,1,2,3,4,6] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => ? = 0
Description
The number of 3-rises of a permutation.
A 3-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+3 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]], for 2-rises see [[St000534]].
Matching statistic: St000709
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000709: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000709: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 1
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => ? = 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0
Description
The number of occurrences of 14-2-3 or 14-3-2.
The number of permutations avoiding both of these patterns is the case $k=2$ of the third item in Corollary 34 of [1].
The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001868The number of alignments of type NE of a signed permutation. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001488The number of corners of a skew partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001061The number of indices that are both descents and recoils of a permutation. St001470The cyclic holeyness of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000234The number of global ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001344The neighbouring number of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001712The number of natural descents of a standard Young tableau. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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 $eA$. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001928The number of non-overlapping descents in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001424The number of distinct squares in a binary word. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000632The jump number of the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000782The indicator function of whether a given perfect matching is an L & P matching. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
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