Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001283
Mapping
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 2
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 1
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 1
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers. A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers. The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
Mapping
St001284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 2
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 1
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 1
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
Description
The number of finite groups that are realised by the given partition over the complex numbers. A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St001371
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 85%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ? => ? = 1
[2]
 => []
 => ?
 => ? => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => => ? ∊ {0,1}
[3]
 => []
 => ?
 => ? => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4]
 => []
 => ?
 => ? => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5]
 => []
 => ?
 => ? => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[2,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6]
 => []
 => ?
 => ? => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[3,3]
 => [3]
 => []
 => => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[2,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[7]
 => []
 => ?
 => ? => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4,3]
 => [3]
 => []
 => => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[3,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[3,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8]
 => []
 => ?
 => ? => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[5,3]
 => [3]
 => []
 => => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[4,4]
 => [4]
 => []
 => => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[4,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,3,2]
 => [3,2]
 => [2]
 => 100 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[9]
 => []
 => ?
 => ? => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[6,3]
 => [3]
 => []
 => => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5,4]
 => [4]
 => []
 => => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[5,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,4,1]
 => [4,1]
 => [1]
 => 10 => 0
[10]
 => []
 => ?
 => ? => ? ∊ {0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => => ? ∊ {0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => => ? ∊ {0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => => ? ∊ {0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => => ? ∊ {0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => => ? ∊ {0,0,1,1,2,3}
[11]
 => []
 => ?
 => ? => ? ∊ {0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => => ? ∊ {0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => => ? ∊ {0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => => ? ∊ {0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => => ? ∊ {0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1111111110 => ? ∊ {0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ? => ? ∊ {1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => => ? ∊ {1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => => ? ∊ {1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => => ? ∊ {1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => => ? ∊ {1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => => ? ∊ {1,1,1,1,1,1,2,2,6}
[6,6]
 => [6]
 => []
 => => ? ∊ {1,1,1,1,1,1,2,2,6}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1111111110 => ? ∊ {1,1,1,1,1,1,2,2,6}
Description
The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001107
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 83%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,6}
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St001695
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 81%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,2,2,6}
Description
The natural comajor index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Matching statistic: St001698
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 81%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,2,2,6}
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 81%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,2,2,6}
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 20% ●values known / values provided: 81%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,2,2,6}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,2,2,6}
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St000687
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 81%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 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]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 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]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2,3}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[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
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2,3}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1}
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,6}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,6}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,6}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,6}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,6}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,2,2,6}
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St001435
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 20% ●values known / values provided: 54%●distinct values known / distinct values provided: 20%
Values
[1]
 => []
 => ?
 => ?
 => ? = 1
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2}
[3,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,2}
[2,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[4,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,0,1}
[3,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,0,1}
[3,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,2}
[5,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {1,1,1,2}
[4,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {1,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[3,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {1,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1}
[6,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1}
[5,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1}
[5,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[4,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1}
[4,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 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
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2,3}
[7,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,0,1,1,2,3}
[6,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,0,1,1,2,3}
[6,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[5,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,0,1,1,2,3}
[5,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[4,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {0,0,1,1,2,3}
[4,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[2],[]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {0,0,1,1,2,3}
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,1,2}
[8,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,1,2}
[7,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,1,2}
[7,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[6,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,1,2}
[6,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,1,2}
[5,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1],[]]
 => 0
[4,3,2]
 => [3,2]
 => [2]
 => [[2],[]]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,1,2}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,1,2}
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[6,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[5,5]
 => [5]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[3,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,2,2,1,1]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,2,1,1,1,1]
 => [2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[10,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[9,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[8,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[7,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
Description
The number of missing boxes in the first row.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000929The constant term of the character polynomial of an integer partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000455The second largest eigenvalue of a graph if it is integral.