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Your data matches 221 different statistics following compositions of up to 3 maps.
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Matching statistic: St000765
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 2 = 1 + 1
[2] => 1 = 0 + 1
[1,1,1] => 3 = 2 + 1
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[3] => 1 = 0 + 1
[1,1,1,1] => 4 = 3 + 1
[1,1,2] => 3 = 2 + 1
[1,2,1] => 2 = 1 + 1
[1,3] => 2 = 1 + 1
[2,1,1] => 1 = 0 + 1
[2,2] => 2 = 1 + 1
[3,1] => 1 = 0 + 1
[4] => 1 = 0 + 1
[1,1,1,1,1] => 5 = 4 + 1
[1,1,1,2] => 4 = 3 + 1
[1,1,2,1] => 3 = 2 + 1
[1,1,3] => 3 = 2 + 1
[1,2,1,1] => 2 = 1 + 1
[1,2,2] => 3 = 2 + 1
[1,3,1] => 2 = 1 + 1
[1,4] => 2 = 1 + 1
[2,1,1,1] => 1 = 0 + 1
[2,1,2] => 2 = 1 + 1
[2,2,1] => 2 = 1 + 1
[2,3] => 2 = 1 + 1
[3,1,1] => 1 = 0 + 1
[3,2] => 1 = 0 + 1
[4,1] => 1 = 0 + 1
[5] => 1 = 0 + 1
[1,1,1,1,1,1] => 6 = 5 + 1
[1,1,1,1,2] => 5 = 4 + 1
[1,1,1,2,1] => 4 = 3 + 1
[1,1,1,3] => 4 = 3 + 1
[1,1,2,1,1] => 3 = 2 + 1
[1,1,2,2] => 4 = 3 + 1
[1,1,3,1] => 3 = 2 + 1
[1,1,4] => 3 = 2 + 1
[1,2,1,1,1] => 2 = 1 + 1
[1,2,1,2] => 3 = 2 + 1
[1,2,2,1] => 3 = 2 + 1
[1,2,3] => 3 = 2 + 1
[1,3,1,1] => 2 = 1 + 1
[1,3,2] => 2 = 1 + 1
[1,4,1] => 2 = 1 + 1
[1,5] => 2 = 1 + 1
[2,1,1,1,1] => 1 = 0 + 1
[2,1,1,2] => 2 = 1 + 1
[2,1,2,1] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
Description
The number of weak records in an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
Matching statistic: St000392
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 11 => 2 = 1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 3 = 2 + 1
[1,2] => 110 => 2 = 1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 4 = 3 + 1
[1,1,2] => 1110 => 3 = 2 + 1
[1,2,1] => 1101 => 2 = 1 + 1
[1,3] => 1100 => 2 = 1 + 1
[2,1,1] => 1011 => 2 = 1 + 1
[2,2] => 1010 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 5 = 4 + 1
[1,1,1,2] => 11110 => 4 = 3 + 1
[1,1,2,1] => 11101 => 3 = 2 + 1
[1,1,3] => 11100 => 3 = 2 + 1
[1,2,1,1] => 11011 => 2 = 1 + 1
[1,2,2] => 11010 => 2 = 1 + 1
[1,3,1] => 11001 => 2 = 1 + 1
[1,4] => 11000 => 2 = 1 + 1
[2,1,1,1] => 10111 => 3 = 2 + 1
[2,1,2] => 10110 => 2 = 1 + 1
[2,2,1] => 10101 => 1 = 0 + 1
[2,3] => 10100 => 1 = 0 + 1
[3,1,1] => 10011 => 2 = 1 + 1
[3,2] => 10010 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 6 = 5 + 1
[1,1,1,1,2] => 111110 => 5 = 4 + 1
[1,1,1,2,1] => 111101 => 4 = 3 + 1
[1,1,1,3] => 111100 => 4 = 3 + 1
[1,1,2,1,1] => 111011 => 3 = 2 + 1
[1,1,2,2] => 111010 => 3 = 2 + 1
[1,1,3,1] => 111001 => 3 = 2 + 1
[1,1,4] => 111000 => 3 = 2 + 1
[1,2,1,1,1] => 110111 => 3 = 2 + 1
[1,2,1,2] => 110110 => 2 = 1 + 1
[1,2,2,1] => 110101 => 2 = 1 + 1
[1,2,3] => 110100 => 2 = 1 + 1
[1,3,1,1] => 110011 => 2 = 1 + 1
[1,3,2] => 110010 => 2 = 1 + 1
[1,4,1] => 110001 => 2 = 1 + 1
[1,5] => 110000 => 2 = 1 + 1
[2,1,1,1,1] => 101111 => 4 = 3 + 1
[2,1,1,2] => 101110 => 3 = 2 + 1
[2,1,2,1] => 101101 => 2 = 1 + 1
[2,1,3] => 101100 => 2 = 1 + 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000723
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => ([],2)
=> 2 = 1 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> 1 = 0 + 1
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => ([],3)
=> 3 = 2 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4] => ([],4)
=> 4 = 3 + 1
[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 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4] => ([(3,4)],5)
=> 3 = 2 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[5] => ([],5)
=> 5 = 4 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5] => ([(4,5)],6)
=> 4 = 3 + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[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)
=> 2 = 1 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph.
The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].
Matching statistic: St001024
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001024: Dyck paths ⟶ 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
[2] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001210
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
Description
Gives 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.
Matching statistic: St001733
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001733: Dyck paths ⟶ 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
[2] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
Description
The number of weak left to right maxima of a Dyck path.
A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its
left.
Matching statistic: St000969
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000969: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> 3 = 1 + 2
[2] => [1,1,0,0]
=> 2 = 0 + 2
[1,1,1] => [1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2] => [1,0,1,1,0,0]
=> 3 = 1 + 2
[2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[3] => [1,1,1,0,0,0]
=> 2 = 0 + 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[4] => [1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6 = 4 + 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 4 = 2 + 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 1 + 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 5 + 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 4 + 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 1 + 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3 = 1 + 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
Description
We 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 $c_{n-1}$. Then we calculate the global dimension of that CNakayama algebra.
Matching statistic: St001184
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001202
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001202: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001202: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra.
The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.
Matching statistic: St001481
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
Description
The minimal height of a peak of a Dyck path.
The following 211 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001933The largest multiplicity of a part in an integer partition. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000700The protection number of an ordered tree. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000013The height of a Dyck path. St000381The largest part of an integer composition. St000439The position of the first down step of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000982The length of the longest constant subword. St001062The maximal size of a block of a set partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001330The hat guessing number of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St000117The number of centered tunnels of a Dyck path. St000148The number of odd parts of a partition. St000288The number of ones in a binary word. St000296The length of the symmetric border of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001910The height of the middle non-run of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000308The height of the tree associated to a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001592The maximal number of simple paths between any two different vertices of a graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000260The radius of a connected graph. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000873The aix statistic of a permutation. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001948The number of augmented double ascents of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St000485The length of the longest cycle of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001712The number of natural descents of a standard Young tableau. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000454The largest eigenvalue of a graph if it is integral. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer 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. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001615The number of join prime elements of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000782The indicator function of whether a given perfect matching is an L & P matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001651The Frankl number of a lattice. St000456The monochromatic index of a connected graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000668The least common multiple of the parts of the partition. St000699The toughness times the least common multiple of 1,. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001587Half of the largest even part of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001657The number of twos in an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001961The sum of the greatest common divisors of all pairs of parts. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001267The length of the Lyndon factorization of the binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St001568The smallest positive integer that does not appear twice in the partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000352The Elizalde-Pak rank of a permutation. St001096The size of the overlap set of a permutation. St000054The first entry of the permutation. St000259The diameter of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000663The number of right floats of a permutation. St001060The distinguishing index of a graph. St001556The number of inversions of the third entry of a permutation. St000090The variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000383The last part of an integer composition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition.
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