- FindStat

Your data matches 127 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000220

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1,0]
 => [1,1,0,0]
 => [2,3,1] => 0
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 0
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 0
[3,1]
 => [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
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
[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
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,1,2,3,4,5,6] => 0
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [6,7,8,1,2,3,4,5] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,9,1,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,7,8,1,2,3,4,5,6] => 0
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [10,9,1,2,3,4,5,6,7,8] => 0
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => 0
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [10,9,8,1,2,3,4,5,6,7] => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 0
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [8,7,4,5,6,1,2,3] => 0
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 0
[7,7]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [7,8,9,10,1,2,3,4,5,6] => 0
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => 0
[]
 => []
 => [1,0]
 => [2,1] => 0
[4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => 0
[5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => 0

Description

The number of occurrences of the pattern 132 in a permutation.

Matching statistic: St000454

Mapping

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
St000454: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 53%●distinct values known / distinct values provided: 20%

Values

[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => 1 = 0 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 1 = 0 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 1 = 0 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 1 = 0 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => 1 = 0 + 1
[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)
 => ? = 3 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 5 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => 1 = 0 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 1 = 0 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => 1 = 0 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? = 0 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? = 0 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
 => ? = 0 + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => 1 = 0 + 1
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => ([(5,6)],7)
 => 1 = 0 + 1
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,5,6,7] => ([(5,6)],7)
 => 1 = 0 + 1
[7,7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7,9] => ([(7,8)],9)
 => ? = 0 + 1
[4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,3,5,4,6,7,8] => ([(6,7)],8)
 => ? = 0 + 1
[]
 => []
 => [] => ([],0)
 => ? = 0 + 1
[4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,5,4,6,7,8,9] => ([(7,8)],9)
 => ? = 0 + 1
[5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,6,5,7,8,9,10] => ([(8,9)],10)
 => ? = 0 + 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: St000422

Mapping

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: 20% ●values known / values provided: 53%●distinct values known / distinct values provided: 20%

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)
 => ? = 2 + 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)
 => ? = 3 + 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 2 + 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 5 + 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => 2 = 0 + 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 2 = 0 + 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
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => 2 = 0 + 2
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? = 0 + 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? = 0 + 2
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? = 0 + 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
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
 => ? = 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
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => ([(5,6)],7)
 => 2 = 0 + 2
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,5,6,7] => ([(5,6)],7)
 => 2 = 0 + 2
[7,7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7,9] => ([(7,8)],9)
 => ? = 0 + 2
[4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,3,5,4,6,7,8] => ([(6,7)],8)
 => ? = 0 + 2
[]
 => []
 => [] => ([],0)
 => ? = 0 + 2
[4,4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,5,4,6,7,8,9] => ([(7,8)],9)
 => ? = 0 + 2
[5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,6,5,7,8,9,10] => ([(8,9)],10)
 => ? = 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

Mapping

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: 20% ●values known / values provided: 41%●distinct values known / distinct values provided: 20%

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)
 => ? = 2
[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)
 => ? = 3
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 0
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? = 0
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[8]
 => 100000000 => [1,9] => ([(8,9)],10)
 => ? = 0
[4,4]
 => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,1,1,1,1,1]
 => 111111110 => [1,1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0
[3,3,3]
 => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,2,2,2,2]
 => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[3,3,3,3]
 => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[7,7]
 => 110000000 => [1,1,8] => ([(7,8),(7,9),(8,9)],10)
 => ? = 0
[4,4,4,4]
 => 11110000 => [1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[]
 =>  => [1] => ([],1)
 => ? = 0
[4,4,4,4,4]
 => 111110000 => [1,1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0
[5,5,5,5,5]
 => 1111100000 => [1,1,1,1,1,6] => ([(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 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: St000369

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 34%●distinct values known / distinct values provided: 20%

Values

[1]
 => []
 => ?
 => ?
 => ? = 0
[2]
 => []
 => ?
 => ?
 => ? = 0
[1,1]
 => [1]
 => []
 => []
 => ? = 0
[3]
 => []
 => ?
 => ?
 => ? = 0
[2,1]
 => [1]
 => []
 => []
 => ? = 1
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? = 0
[4]
 => []
 => ?
 => ?
 => ? = 0
[3,1]
 => [1]
 => []
 => []
 => ? = 2
[2,2]
 => [2]
 => []
 => []
 => ? = 0
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? = 2
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? = 0
[4,1]
 => [1]
 => []
 => []
 => ? = 3
[3,2]
 => [2]
 => []
 => []
 => ? = 2
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? = 5
[6]
 => []
 => ?
 => ?
 => ? = 0
[3,3]
 => [3]
 => []
 => []
 => ? = 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? = 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? = 0
[4,4]
 => [4]
 => []
 => []
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,3,3,3]
 => [3,3,3]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
[7,7]
 => [7]
 => []
 => []
 => ? = 0
[4,4,4,4]
 => [4,4,4]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[]
 => ?
 => ?
 => ?
 => ? = 0
[4,4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[5,5,5,5,5]
 => [5,5,5,5]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 0

Description

The dinv deficit of a Dyck path. For a Dyck path

$D$ of semilength

$n$ , this is defined as

$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$ In other words, this is the number of boxes in the partition traced out by

$D$ for which the leg-length minus the arm-length is not in

$\{0,1\}$ . See also [[St000376]] for the bounce deficit.

Matching statistic: St000791

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 34%●distinct values known / distinct values provided: 20%

Values

[1]
 => []
 => ?
 => ?
 => ? = 0
[2]
 => []
 => ?
 => ?
 => ? = 0
[1,1]
 => [1]
 => []
 => []
 => ? = 0
[3]
 => []
 => ?
 => ?
 => ? = 0
[2,1]
 => [1]
 => []
 => []
 => ? = 1
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? = 0
[4]
 => []
 => ?
 => ?
 => ? = 0
[3,1]
 => [1]
 => []
 => []
 => ? = 2
[2,2]
 => [2]
 => []
 => []
 => ? = 0
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? = 2
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? = 0
[4,1]
 => [1]
 => []
 => []
 => ? = 3
[3,2]
 => [2]
 => []
 => []
 => ? = 2
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? = 5
[6]
 => []
 => ?
 => ?
 => ? = 0
[3,3]
 => [3]
 => []
 => []
 => ? = 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? = 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? = 0
[4,4]
 => [4]
 => []
 => []
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,3,3,3]
 => [3,3,3]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
[7,7]
 => [7]
 => []
 => []
 => ? = 0
[4,4,4,4]
 => [4,4,4]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[]
 => ?
 => ?
 => ?
 => ? = 0
[4,4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[5,5,5,5,5]
 => [5,5,5,5]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 0

Description

The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].

Matching statistic: St001141

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 34%●distinct values known / distinct values provided: 20%

Values

[1]
 => []
 => ?
 => ?
 => ? = 0
[2]
 => []
 => ?
 => ?
 => ? = 0
[1,1]
 => [1]
 => []
 => []
 => ? = 0
[3]
 => []
 => ?
 => ?
 => ? = 0
[2,1]
 => [1]
 => []
 => []
 => ? = 1
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? = 0
[4]
 => []
 => ?
 => ?
 => ? = 0
[3,1]
 => [1]
 => []
 => []
 => ? = 2
[2,2]
 => [2]
 => []
 => []
 => ? = 0
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? = 2
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? = 0
[4,1]
 => [1]
 => []
 => []
 => ? = 3
[3,2]
 => [2]
 => []
 => []
 => ? = 2
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? = 5
[6]
 => []
 => ?
 => ?
 => ? = 0
[3,3]
 => [3]
 => []
 => []
 => ? = 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? = 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? = 0
[4,4]
 => [4]
 => []
 => []
 => ? = 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,3,3,3]
 => [3,3,3]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 0
[7,7]
 => [7]
 => []
 => []
 => ? = 0
[4,4,4,4]
 => [4,4,4]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[]
 => ?
 => ?
 => ?
 => ? = 0
[4,4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[5,5,5,5,5]
 => [5,5,5,5]
 => [5,5,5]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 0

Description

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

Matching statistic: St000068
(load all 3 compositions to match this statistic)

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%

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)
 => ? = 2 + 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)
 => ? = 3 + 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)
 => ? = 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)
 => ? = 5 + 1
[6]
 => 1000000 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
 => ? = 0 + 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
[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
[1,1,1,1,1,1]
 => 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
 => ? = 0 + 1
[7]
 => 10000000 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
 => ? = 0 + 1
[1,1,1,1,1,1,1]
 => 11111110 => ([(0,2),(0,8),(1,10),(2,9),(3,5),(3,11),(4,3),(4,13),(5,7),(5,12),(6,4),(6,15),(7,1),(7,14),(8,6),(8,9),(9,15),(11,12),(12,14),(13,11),(14,10),(15,13)],16)
 => ? = 0 + 1
[8]
 => 100000000 => ([(0,2),(0,9),(1,11),(2,10),(3,4),(3,13),(4,6),(4,12),(5,3),(5,15),(6,8),(6,14),(7,5),(7,17),(8,1),(8,16),(9,7),(9,10),(10,17),(12,14),(13,12),(14,16),(15,13),(16,11),(17,15)],18)
 => ? = 0 + 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
[1,1,1,1,1,1,1,1]
 => 111111110 => ([(0,2),(0,9),(1,11),(2,10),(3,4),(3,13),(4,6),(4,12),(5,3),(5,15),(6,8),(6,14),(7,5),(7,17),(8,1),(8,16),(9,7),(9,10),(10,17),(12,14),(13,12),(14,16),(15,13),(16,11),(17,15)],18)
 => ? = 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
[2,2,2,2,2]
 => 1111100 => ([(0,6),(0,7),(1,11),(2,5),(2,15),(3,13),(4,3),(4,17),(5,4),(5,16),(6,2),(6,14),(7,1),(7,14),(9,12),(10,9),(11,10),(12,8),(13,8),(14,11),(14,15),(15,10),(15,16),(16,9),(16,17),(17,12),(17,13)],18)
 => ? = 0 + 1
[3,3,3,3]
 => 1111000 => ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
 => ? = 0 + 1
[7,7]
 => 110000000 => ([(0,8),(0,9),(1,12),(2,4),(2,18),(3,17),(4,6),(4,20),(5,7),(5,22),(6,5),(6,23),(7,3),(7,21),(8,2),(8,19),(9,1),(9,19),(11,14),(12,11),(13,15),(14,13),(15,16),(16,10),(17,10),(18,11),(18,20),(19,12),(19,18),(20,14),(20,23),(21,16),(21,17),(22,15),(22,21),(23,13),(23,22)],24)
 => ? = 0 + 1
[4,4,4,4]
 => 11110000 => ([(0,7),(0,8),(1,6),(1,19),(2,5),(2,18),(3,14),(4,15),(5,3),(5,23),(6,4),(6,24),(7,2),(7,20),(8,1),(8,20),(10,12),(11,13),(12,9),(13,9),(14,10),(15,11),(16,21),(16,22),(17,12),(17,13),(18,16),(18,23),(19,16),(19,24),(20,18),(20,19),(21,10),(21,17),(22,11),(22,17),(23,14),(23,21),(24,15),(24,22)],25)
 => ? = 0 + 1
[]
 =>  => ?
 => ? = 0 + 1
[4,4,4,4,4]
 => 111110000 => ([(0,8),(0,9),(1,7),(1,23),(2,6),(2,22),(3,17),(4,16),(5,4),(5,29),(6,5),(6,27),(7,3),(7,28),(8,1),(8,24),(9,2),(9,24),(11,15),(12,13),(13,14),(14,10),(15,10),(16,11),(17,12),(18,20),(18,26),(19,18),(19,25),(20,13),(20,21),(21,14),(21,15),(22,19),(22,27),(23,19),(23,28),(24,22),(24,23),(25,12),(25,20),(26,11),(26,21),(27,18),(27,29),(28,17),(28,25),(29,16),(29,26)],30)
 => ? = 0 + 1
[5,5,5,5,5]
 => 1111100000 => ([(0,9),(0,10),(1,8),(1,28),(2,7),(2,27),(3,18),(4,19),(5,3),(5,34),(6,4),(6,35),(7,5),(7,32),(8,6),(8,33),(9,2),(9,29),(10,1),(10,29),(12,14),(13,15),(14,16),(15,17),(16,11),(17,11),(18,12),(19,13),(20,22),(20,23),(21,25),(21,26),(22,14),(22,24),(23,15),(23,24),(24,16),(24,17),(25,20),(25,30),(26,20),(26,31),(27,21),(27,32),(28,21),(28,33),(29,27),(29,28),(30,12),(30,22),(31,13),(31,23),(32,25),(32,34),(33,26),(33,35),(34,18),(34,30),(35,19),(35,31)],36)
 => ? = 0 + 1

Description

The number of minimal elements in a poset.

Matching statistic: St000704
(load all 4 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%

Values

[1]
 => []
 => ?
 => ? = 0 + 1
[2]
 => []
 => ?
 => ? = 0 + 1
[1,1]
 => [1]
 => []
 => ? = 0 + 1
[3]
 => []
 => ?
 => ? = 0 + 1
[2,1]
 => [1]
 => []
 => ? = 1 + 1
[1,1,1]
 => [1,1]
 => [1]
 => ? = 0 + 1
[4]
 => []
 => ?
 => ? = 0 + 1
[3,1]
 => [1]
 => []
 => ? = 2 + 1
[2,2]
 => [2]
 => []
 => ? = 0 + 1
[2,1,1]
 => [1,1]
 => [1]
 => ? = 2 + 1
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[5]
 => []
 => ?
 => ? = 0 + 1
[4,1]
 => [1]
 => []
 => ? = 3 + 1
[3,2]
 => [2]
 => []
 => ? = 2 + 1
[2,2,1]
 => [2,1]
 => [1]
 => ? = 5 + 1
[6]
 => []
 => ?
 => ? = 0 + 1
[3,3]
 => [3]
 => []
 => ? = 0 + 1
[2,2,2]
 => [2,2]
 => [2]
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
[7]
 => []
 => ?
 => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
[8]
 => []
 => ?
 => ? = 0 + 1
[4,4]
 => [4]
 => []
 => ? = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[3,3,3]
 => [3,3]
 => [3]
 => 1 = 0 + 1
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => 1 = 0 + 1
[3,3,3,3]
 => [3,3,3]
 => [3,3]
 => 1 = 0 + 1
[7,7]
 => [7]
 => []
 => ? = 0 + 1
[4,4,4,4]
 => [4,4,4]
 => [4,4]
 => 1 = 0 + 1
[]
 => ?
 => ?
 => ? = 0 + 1
[4,4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => 1 = 0 + 1
[5,5,5,5,5]
 => [5,5,5,5]
 => [5,5,5]
 => ? = 0 + 1

Description

The number of semistandard tableaux on a given integer partition with minimal maximal entry. This is, for an integer partition

$\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$ , the number of [[SemistandardTableaux|semistandard tableaux]] of shape

$\lambda$ with maximal entry

$k$ . Equivalently, this is the evaluation

$s_\lambda(1,\ldots,1)$ of the Schur function

$s_\lambda$ in

$k$ variables, or, explicitly,

$\prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$ where the product is over all cells

$(i,j) \in L$ and

$\operatorname{hook}(i,j)$ is the hook length of a cell. See [Theorem 6.3, 1] for details.

Matching statistic: St001128
(load all 4 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%

Values

[1]
 => []
 => ?
 => ? = 0 + 1
[2]
 => []
 => ?
 => ? = 0 + 1
[1,1]
 => [1]
 => []
 => ? = 0 + 1
[3]
 => []
 => ?
 => ? = 0 + 1
[2,1]
 => [1]
 => []
 => ? = 1 + 1
[1,1,1]
 => [1,1]
 => [1]
 => ? = 0 + 1
[4]
 => []
 => ?
 => ? = 0 + 1
[3,1]
 => [1]
 => []
 => ? = 2 + 1
[2,2]
 => [2]
 => []
 => ? = 0 + 1
[2,1,1]
 => [1,1]
 => [1]
 => ? = 2 + 1
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[5]
 => []
 => ?
 => ? = 0 + 1
[4,1]
 => [1]
 => []
 => ? = 3 + 1
[3,2]
 => [2]
 => []
 => ? = 2 + 1
[2,2,1]
 => [2,1]
 => [1]
 => ? = 5 + 1
[6]
 => []
 => ?
 => ? = 0 + 1
[3,3]
 => [3]
 => []
 => ? = 0 + 1
[2,2,2]
 => [2,2]
 => [2]
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
[7]
 => []
 => ?
 => ? = 0 + 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
[8]
 => []
 => ?
 => ? = 0 + 1
[4,4]
 => [4]
 => []
 => ? = 0 + 1
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[3,3,3]
 => [3,3]
 => [3]
 => 1 = 0 + 1
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => 1 = 0 + 1
[3,3,3,3]
 => [3,3,3]
 => [3,3]
 => 1 = 0 + 1
[7,7]
 => [7]
 => []
 => ? = 0 + 1
[4,4,4,4]
 => [4,4,4]
 => [4,4]
 => 1 = 0 + 1
[]
 => ?
 => ?
 => ? = 0 + 1
[4,4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => 1 = 0 + 1
[5,5,5,5,5]
 => [5,5,5,5]
 => [5,5,5]
 => ? = 0 + 1

Description

The exponens consonantiae of a partition. This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].

The following 117 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001811The Castelnuovo-Mumford regularity of a permutation. St001868The number of alignments of type NE of a signed permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001487The number of inner corners of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000090The variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001722The number of minimal chains with small intervals between a binary word and the top element. St000264The girth of a graph, which is not a tree. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000871The number of very big ascents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. 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. 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

$eA$ . 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

$eA$ . 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

$eA$ . 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001928The number of non-overlapping descents in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. 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

$K[x]/(x^n)$ . St001569The maximal modular displacement of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type.