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Your data matches 826 different statistics following compositions of up to 3 maps.
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Matching statistic: St000480
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 1
[2,4,3,1] => [4,3,2,1] => [2,2]
=> 1
[4,3,1,2] => [4,3,2,1] => [2,2]
=> 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> 1
[2,5,4,1,3] => [4,5,3,1,2] => [2,2,1]
=> 1
[2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[3,1,2,5,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[3,1,5,4,2] => [5,2,4,3,1] => [2,2,1]
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[3,2,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> 1
[3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
=> 1
[3,5,4,2,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,1,2,3] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,1,3,2] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,2,1,3] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,2,3,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,3,1,2] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,1,1,1,1]
=> 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,1,1,1,1]
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,1,1]
=> 1
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> 1
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [2,2,1,1]
=> 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,2,1,1]
=> 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [2,2,1,1]
=> 1
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [2,2,1,1]
=> 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [2,2,1,1]
=> 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> 1
[1,4,2,6,5,3] => [1,6,3,5,4,2] => [2,2,1,1]
=> 1
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> 1
[1,4,3,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St001092
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 1
[2,4,3,1] => [4,3,2,1] => [2,2]
=> 1
[4,3,1,2] => [4,3,2,1] => [2,2]
=> 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> 1
[2,5,4,1,3] => [4,5,3,1,2] => [2,2,1]
=> 1
[2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[3,1,2,5,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[3,1,5,4,2] => [5,2,4,3,1] => [2,2,1]
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 1
[3,2,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> 1
[3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
=> 1
[3,5,4,2,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,1,2,3] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,1,3,2] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,2,1,3] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,2,3,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,3,1,2] => [5,4,3,2,1] => [2,2,1]
=> 1
[5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,1,1,1,1]
=> 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,1,1,1,1]
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,1,1]
=> 1
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> 1
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [2,2,1,1]
=> 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,2,1,1]
=> 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [2,2,1,1]
=> 1
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [2,2,1,1]
=> 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [2,2,1,1]
=> 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> 1
[1,4,2,6,5,3] => [1,6,3,5,4,2] => [2,2,1,1]
=> 1
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> 1
[1,4,3,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000260
(load all 105 compositions to match this statistic)
(load all 105 compositions to match this statistic)
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => ([],1)
=> 0
[2,1] => 1 => [1,1] => ([(0,1)],2)
=> 1
[1,3,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,4,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,5,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,3,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3,5,4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,5,4,3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,1,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,5,4,1] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,1,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,3,1] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,4,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,1,5,4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,4,1] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,4,1,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,4,2,1] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,2,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,3,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,1,3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,3,1] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,1,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,2,1] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,4,6,5] => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,2,3,6,5,4] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,3,6,5] => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,6,5,3] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,6,5,3,4] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,6,5,4,3] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2,4,6,5] => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2,6,5,4] => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,4,2,6,5] => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,4,6,5,2] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,6,5,2,4] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,6,5,4,2] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,2,3,6,5] => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,2,6,5,3] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,3,2,6,5] => 01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,3,6,5,2] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000374
(load all 253 compositions to match this statistic)
(load all 253 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 0
[2,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[2,5,4,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[5,4,1,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,2,3,4,6,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1
[1,2,3,6,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[1,2,4,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,2,4,6,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[1,2,6,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 1
[1,3,2,4,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,3,2,6,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,4,2,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,3,4,6,5,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[1,3,6,5,2,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 1
[1,4,2,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,4,2,6,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,3,2,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,3,6,5,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000481
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> [1,1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1
[3,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2,2]
=> 1
[2,4,3,1] => [4,3,2,1] => [2,2]
=> [2,2]
=> 1
[4,3,1,2] => [4,3,2,1] => [2,2]
=> [2,2]
=> 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2,2]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [3,2]
=> 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 1
[2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> [3,2]
=> 1
[2,5,4,1,3] => [4,5,3,1,2] => [2,2,1]
=> [3,2]
=> 1
[2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[3,1,2,5,4] => [3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 1
[3,1,5,4,2] => [5,2,4,3,1] => [2,2,1]
=> [3,2]
=> 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 1
[3,2,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> [3,2]
=> 1
[3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[3,5,4,2,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[5,4,1,2,3] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[5,4,1,3,2] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[5,4,2,1,3] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[5,4,2,3,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[5,4,3,1,2] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> [4,2]
=> 1
[1,4,2,6,5,3] => [1,6,3,5,4,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> [4,2]
=> 1
[1,4,3,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> [4,2]
=> 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000742
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[2,1] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,2,4,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[2,4,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,2,3,5,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,3,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,4,2,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,5,4,3,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,3,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[2,1,5,4,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,3,1,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[2,3,5,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[2,5,4,1,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,5,4,3,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[3,1,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[3,1,5,4,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,2,1,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,2,5,4,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,5,4,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,5,4,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[5,4,1,2,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[5,4,1,3,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[5,4,2,1,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[5,4,2,3,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[5,4,3,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,2,3,4,6,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
[1,2,3,6,5,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,2,4,3,6,5] => [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
[1,2,4,6,5,3] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,2,6,5,3,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,2,6,5,4,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[1,3,2,4,6,5] => [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
[1,3,2,6,5,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,3,4,2,6,5] => [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
[1,3,4,6,5,2] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,3,6,5,2,4] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,3,6,5,4,2] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[1,4,2,3,6,5] => [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
[1,4,2,6,5,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,4,3,2,6,5] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,4,3,6,5,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
Description
The number of big ascents of a permutation after prepending zero.
Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$.
It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]).
G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]].
[[St000646]] is the statistic without the border condition $\pi(0) = 0$.
Matching statistic: St001271
(load all 86 compositions to match this statistic)
(load all 86 compositions to match this statistic)
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => ([],1)
=> 0
[2,1] => 1 => [1,1] => ([(0,1)],2)
=> 1
[1,3,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,4,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,5,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,3,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,3,5,4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,5,4,3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,1,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,5,4,1] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,1,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,4,3,1] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,4,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,1,5,4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,4,1] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,4,1,2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,5,4,2,1] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,2,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,3,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,1,3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,3,1] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,1,2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,2,1] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,4,6,5] => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,2,3,6,5,4] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,3,6,5] => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,6,5,3] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,6,5,3,4] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,6,5,4,3] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2,4,6,5] => 01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,2,6,5,4] => 01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,4,2,6,5] => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,4,6,5,2] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,6,5,2,4] => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,6,5,4,2] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,2,3,6,5] => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,2,6,5,3] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,3,2,6,5] => 01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,4,3,6,5,2] => 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000346
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1 = 0 + 1
[2,1] => [2,1] => [2]
=> [1,1]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2,2]
=> 2 = 1 + 1
[2,4,3,1] => [4,3,2,1] => [2,2]
=> [2,2]
=> 2 = 1 + 1
[4,3,1,2] => [4,3,2,1] => [2,2]
=> [2,2]
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2,2]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,5,4,1,3] => [4,5,3,1,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[5,4,1,2,3] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[5,4,1,3,2] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[5,4,2,1,3] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[5,4,2,3,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[5,4,3,1,2] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,1,1,1,1]
=> [5,1]
=> 2 = 1 + 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,1,1,1,1]
=> [5,1]
=> 2 = 1 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,4,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,6,5,2,4] => [1,5,6,4,2,3] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,4,2,6,5,3] => [1,6,3,5,4,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,4,3,6,5,2] => [1,6,3,5,4,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
Description
The number of coarsenings of a partition.
A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000451
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 1 = 0 + 1
[2,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[2,5,4,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[5,4,1,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[1,2,3,4,6,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 1 + 1
[1,2,3,6,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2 = 1 + 1
[1,2,4,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,2,4,6,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2 = 1 + 1
[1,2,6,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2 = 1 + 1
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2 = 1 + 1
[1,3,2,4,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,3,2,6,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
[1,3,4,2,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,3,4,6,5,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2 = 1 + 1
[1,3,6,5,2,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2 = 1 + 1
[1,4,2,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,4,2,6,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
[1,4,3,2,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
[1,4,3,6,5,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000920
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,5,4,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[5,4,1,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,6,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,6,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,6,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,6,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,6,5,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,6,5,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,6,5,2,4] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,6,5,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,6,5] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,6,5,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
The following 816 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000007The number of saliances of the permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000535The rank-width of a graph. St000845The maximal number of elements covered by an element in a poset. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001587Half of the largest even part of an integer partition. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St000025The number of initial rises of a Dyck path. St000053The number of valleys of the Dyck path. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000143The largest repeated part of a partition. St000183The side length of the Durfee square of an integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000306The bounce count of a Dyck path. St000536The pathwidth of a graph. St000651The maximal size of a rise in a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000846The maximal number of elements covering an element of a poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001395The number of strictly unfriendly partitions of a graph. St001484The number of singletons of an integer partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001512The minimum rank of a graph. St001613The binary logarithm of the size of the center of a lattice. St001777The number of weak descents in an integer composition. St001826The maximal number of leaves on a vertex of a graph. St001931The weak major index of an integer composition regarded as a word. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000058The order of a permutation. St000147The largest part of an integer partition. St000258The burning number of a graph. St000273The domination number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000397The Strahler number of a rooted tree. St000453The number of distinct Laplacian eigenvalues of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001654The monophonic hull number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001829The common independence number of a graph. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000678The number of up steps after the last double rise of a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000629The defect of a binary word. St000929The constant term of the character polynomial of an integer partition. St000054The first entry of the permutation. St000253The crossing number of a set partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000456The monochromatic index of a connected graph. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000729The minimal arc length of a set partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000823The number of unsplittable factors of the set partition. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001468The smallest fixpoint of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000210Minimum over maximum difference of elements in cycles. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000352The Elizalde-Pak rank of a permutation. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000478Another weight of a partition according to Alladi. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000668The least common multiple of the parts of the partition. St000674The number of hills of a Dyck path. St000699The toughness times the least common multiple of 1,. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000842The breadth of a permutation. St000921The number of internal inversions of a binary word. St000962The 3-shifted major index of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. 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. St001281The normalized isoperimetric number of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001793The difference between the clique number and the chromatic number of a graph. St000026The position of the first return of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000056The decomposition (or block) number of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000096The number of spanning trees of a graph. St000115The single entry in the last row. St000254The nesting number of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000505The biggest entry in the block containing the 1. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000544The cop number of a graph. St000553The number of blocks of a graph. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000701The protection number of a binary tree. St000706The product of the factorials of the multiplicities of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000730The maximal arc length of a set partition. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000908The length of the shortest maximal antichain in a poset. St000913The number of ways to refine the partition into singletons. St000916The packing number of a graph. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000948The chromatic discriminant of a graph. St000983The length of the longest alternating subword. St000991The number of right-to-left minima of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001260The permanent of an alternating sign matrix. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001463The number of distinct columns in the nullspace of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001597The Frobenius rank of a skew partition. St001642The Prague dimension of a graph. St001665The number of pure excedances of a permutation. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. 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. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001828The Euler characteristic of a graph. St001838The number of nonempty primitive factors of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000095The number of triangles of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000133The "bounce" of a permutation. 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. St000217The number of occurrences of the pattern 312 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000232The number of crossings of a set partition. St000234The number of global ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000351The determinant of the adjacency matrix of a graph. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000376The bounce deficit of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000391The sum of the positions of the ones in a binary word. St000403The Szeged index minus the Wiener index of a graph. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000444The length of the maximal rise of a Dyck path. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000502The number of successions of a set partitions. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000516The number of stretching pairs of a permutation. St000552The number of cut vertices of a graph. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000628The balance of a binary word. St000637The length of the longest cycle in a graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000661The number of rises of length 3 of a Dyck path. St000671The maximin edge-connectivity for choosing a subgraph. St000682The Grundy value of Welter's game on a binary word. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000691The number of changes of a binary word. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000753The Grundy value for the game of Kayles on a binary word. St000768The number of peaks in an integer composition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000963The 2-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000974The length of the trunk of an ordered tree. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001301The first Betti number of the order complex associated with the poset. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001357The maximal degree of a regular spanning subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001374The Padmakar-Ivan index of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001584The area statistic between a Dyck path and its bounce path. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001657The number of twos in an integer partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001692The number of vertices with higher degree than the average degree in a graph. 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. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001708The number of pairs of vertices of different degree in a graph. St001712The number of natural descents of a standard Young tableau. St001718The number of non-empty open intervals in a poset. St001736The total number of cycles in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001845The number of join irreducibles minus the rank of a lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000989The number of final rises of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000061The number of nodes on the left branch of a binary tree. St000354The number of recoils of a permutation. St000570The Edelman-Greene number of a permutation. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000914The sum of the values of the Möbius function of a poset. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001393The induced matching number of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000461The rix statistic of a permutation. St000485The length of the longest cycle of a permutation. St000649The number of 3-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000873The aix statistic of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001261The Castelnuovo-Mumford regularity of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000264The girth of a graph, which is not a tree. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000068The number of minimal elements in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001741The largest integer such that all patterns of this size are contained in the permutation. St000010The length of the partition. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. 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. St000759The smallest missing part in an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001780The order of promotion on the set of standard tableaux of given shape. St001890The maximum magnitude of the Möbius function of a poset. St000209Maximum difference of elements in cycles. St000662The staircase size of the code of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000886The number of permutations with the same antidiagonal sums. St000956The maximal displacement of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000889The number of alternating sign matrices with the same antidiagonal sums. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000834The number of right outer peaks of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000225Difference between largest and smallest parts in a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001947The number of ties in a parking function. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000035The number of left outer peaks of a permutation. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000396The register function (or Horton-Strahler number) of a binary tree. St000650The number of 3-rises of a permutation. St000477The weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001651The Frankl number of a lattice. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the 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. St000284The Plancherel distribution on integer partitions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000567The sum of the products of all pairs of parts. 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. 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. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000961The shifted major index of a permutation. St000883The number of longest increasing subsequences of a permutation. St000245The number of ascents of a permutation. St001462The number of factors of a standard tableaux under concatenation. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000031The number of cycles in the cycle decomposition of a permutation. St000862The number of parts of the shifted shape of a permutation. St001737The number of descents of type 2 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000783The side length of the largest staircase partition fitting into a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000822The Hadwiger number of the graph. St001060The distinguishing index of a graph. St000652The maximal difference between successive positions of a permutation. St000462The major index minus the number of excedences of a permutation. St000646The number of big ascents of a permutation. St001096The size of the overlap set of a permutation. St001052The length of the exterior of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000741The Colin de Verdière graph invariant. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001461The number of topologically connected components of the chord diagram of a permutation. St000141The maximum drop size of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001875The number of simple modules with projective dimension at most 1. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001111The weak 2-dynamic chromatic number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001716The 1-improper chromatic number of a graph. St000062The length of the longest increasing subsequence of the permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000331The number of upper interactions of a Dyck path. St000647The number of big descents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001644The dimension of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000271The chromatic index of a graph. St001108The 2-dynamic chromatic number of a graph. St001656The monophonic position number of a graph. 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 $K[x]/(x^n)$. St001429The number of negative entries in a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St000060The greater neighbor of the maximum. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000472The sum of the ascent bottoms of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001220The width of a permutation. 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. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001520The number of strict 3-descents. St000219The number of occurrences of the pattern 231 in a permutation. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000850The number of 1/2-balanced pairs in a poset. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001964The interval resolution global dimension of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. 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 $eA$. St001638The book thickness of a graph. 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. St001116The game chromatic number of a graph. St000100The number of linear extensions of a poset. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001545The second Elser number of a connected graph. St001624The breadth of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001625The Möbius invariant of a lattice. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000624The normalized sum of the minimal distances to a greater element. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001768The number of reduced words of a signed permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001556The number of inversions of the third entry of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001555The order of a signed permutation. St000635The number of strictly order preserving maps of a poset into itself. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St000454The largest eigenvalue of a graph if it is integral.
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