Your data matches 119 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001501
(load all 48 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001501: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 8
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
Description
The dominant dimension of magnitude 1 Nakayama algebras. We use the code below to biject them to Dyck paths.
Matching statistic: St000207
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? = 2
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? = 4
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 6
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 8
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 12
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? = 2
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? = 4
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 6
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 8
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 12
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000618
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [1,1]
 => [1]
 => 1
[1,1,0,0]
 => [1,2] => [2]
 => []
 => ? = 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [3]
 => []
 => ? = 4
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,1]
 => [1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4]
 => []
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [3,2]
 => [2]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [3,2]
 => [2]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [5]
 => []
 => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [6]
 => []
 => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [7]
 => []
 => ? = 12
Description
The number of self-evacuating tableaux of given shape. This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000655
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => []
 => []
 => []
 => ? = 2
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 4
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 12
Description
The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].
Matching statistic: St000667
(load all 3 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? = 2
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? = 4
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 6
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 8
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 12
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000755
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? = 2
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? = 4
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 6
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 8
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 12
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St000781
(load all 11 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [1,1]
 => [1]
 => 1
[1,1,0,0]
 => [1,2] => [2]
 => []
 => ? = 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [3]
 => []
 => ? = 4
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,1]
 => [1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4]
 => []
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [3,2]
 => [2]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [3,2]
 => [2]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [3,2]
 => [2]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,2]
 => [2]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [3,2]
 => [2]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [5]
 => []
 => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [6]
 => []
 => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [7]
 => []
 => ? = 12
Description
The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001199
(load all 17 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => ? = 6
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 12
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001389
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? = 2
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? = 4
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? = 6
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? = 8
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7]
 => []
 => ? = 12
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
The following 109 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000326The position of the first one in a binary word after appending a 1 at the end. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000047The number of standard immaculate tableaux of a given shape. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St000627The exponent of a binary word. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000260The radius of a connected graph. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000701The protection number of a binary tree. St000255The number of reduced Kogan faces with the permutation as type. St001256Number of simple reflexive modules that are 2-stable reflexive. St000056The decomposition (or block) number of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001732The number of peaks visible from the left. St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000914The sum of the values of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000764The number of strong records in an integer composition. St001890The maximum magnitude of the Möbius function of a poset. St000763The sum of the positions of the strong records of an integer composition. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000788The number of nesting-similar perfect matchings of a perfect matching. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $K[x]/(x^n)$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St001877Number of indecomposable injective modules with projective dimension 2. St000181The number of connected components of the Hasse diagram for the poset. St000669The number of permutations obtained by switching ascents or descents of size 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph.