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Your data matches 141 different statistics following compositions of up to 3 maps.
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Matching statistic: St001501
(load all 176 compositions to match this statistic)
(load all 176 compositions to match this statistic)
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%
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
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%
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
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%
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)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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%
Mp00204: Permutations —LLPS⟶ 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] => [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 self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000667
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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%
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
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%
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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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%
Mp00204: Permutations —LLPS⟶ 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] => [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 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: St001389
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%
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
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001432
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: 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 order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001442
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 99%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,2] => [2]
=> []
=> ? = 2
[1,1,0,0]
=> [2,1] => [1,1]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> []
=> ? = 4
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> []
=> ? = 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? = 8
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [6]
=> []
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7]
=> []
=> ? = 12
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. 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. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000629The defect of a binary word. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000993The multiplicity of the largest part of an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000326The position of the first one in a binary word after appending a 1 at the end. St000296The length of the symmetric border of a binary word. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001371The length of the longest Yamanouchi prefix of a binary word. St000455The second largest eigenvalue of a graph if it is integral. St001256Number of simple reflexive modules that are 2-stable reflexive. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001060The distinguishing index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000259The diameter of a connected graph. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. 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. St000379The number of Hamiltonian cycles in a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant.
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