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Your data matches 127 different statistics following compositions of up to 3 maps.
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Matching statistic: St001335
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0
1 => [1] => [1,0]
=> [1,0]
=> ? = 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> ? = 0
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> ? = 0
000 => [3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001878
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 1
0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1
1000 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
1110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001876
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 - 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 - 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0 - 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0 - 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
1000 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
1110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 - 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 - 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0 - 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0 - 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
1000 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
1110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 1 - 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001198
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
=> ? = 0 + 1
1 => [1] => [1,0]
=> ? = 0 + 1
00 => [2] => [1,1,0,0]
=> ? = 0 + 1
01 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
10 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
11 => [2] => [1,1,0,0]
=> ? = 0 + 1
000 => [3] => [1,1,1,0,0,0]
=> ? = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
=> ? = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
Description
The 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 $eA$.
Matching statistic: St001206
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
=> ? = 0 + 1
1 => [1] => [1,0]
=> ? = 0 + 1
00 => [2] => [1,1,0,0]
=> ? = 0 + 1
01 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
10 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
11 => [2] => [1,1,0,0]
=> ? = 0 + 1
000 => [3] => [1,1,1,0,0,0]
=> ? = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
110 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
=> ? = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
Description
The 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 $eA$.
Matching statistic: St000207
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1]
=> []
=> ? = 0
1 => [1] => [1]
=> []
=> ? = 0
00 => [2] => [2]
=> []
=> ? = 0
01 => [1,1] => [1,1]
=> [1]
=> 1
10 => [1,1] => [1,1]
=> [1]
=> 1
11 => [2] => [2]
=> []
=> ? = 0
000 => [3] => [3]
=> []
=> ? = 0
001 => [2,1] => [2,1]
=> [1]
=> 1
010 => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
011 => [1,2] => [2,1]
=> [1]
=> 1
100 => [1,2] => [2,1]
=> [1]
=> 1
101 => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
110 => [2,1] => [2,1]
=> [1]
=> 1
111 => [3] => [3]
=> []
=> ? = 0
0000 => [4] => [4]
=> []
=> ? = 1
0001 => [3,1] => [3,1]
=> [1]
=> 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
0111 => [1,3] => [3,1]
=> [1]
=> 1
1000 => [1,3] => [3,1]
=> [1]
=> 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
1110 => [3,1] => [3,1]
=> [1]
=> 1
1111 => [4] => [4]
=> []
=> ? = 1
00000 => [5] => [5]
=> []
=> ? = 1
11111 => [5] => [5]
=> []
=> ? = 1
000000 => [6] => [6]
=> []
=> ? = 1
111111 => [6] => [6]
=> []
=> ? = 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1]
=> []
=> ? = 0
1 => [1] => [1]
=> []
=> ? = 0
00 => [2] => [2]
=> []
=> ? = 0
01 => [1,1] => [1,1]
=> [1]
=> 1
10 => [1,1] => [1,1]
=> [1]
=> 1
11 => [2] => [2]
=> []
=> ? = 0
000 => [3] => [3]
=> []
=> ? = 0
001 => [2,1] => [2,1]
=> [1]
=> 1
010 => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
011 => [1,2] => [2,1]
=> [1]
=> 1
100 => [1,2] => [2,1]
=> [1]
=> 1
101 => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
110 => [2,1] => [2,1]
=> [1]
=> 1
111 => [3] => [3]
=> []
=> ? = 0
0000 => [4] => [4]
=> []
=> ? = 1
0001 => [3,1] => [3,1]
=> [1]
=> 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
0111 => [1,3] => [3,1]
=> [1]
=> 1
1000 => [1,3] => [3,1]
=> [1]
=> 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
1110 => [3,1] => [3,1]
=> [1]
=> 1
1111 => [4] => [4]
=> []
=> ? = 1
00000 => [5] => [5]
=> []
=> ? = 1
11111 => [5] => [5]
=> []
=> ? = 1
000000 => [6] => [6]
=> []
=> ? = 1
111111 => [6] => [6]
=> []
=> ? = 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000260
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => ([],1)
=> 0
1 => 1 => [1] => ([],1)
=> 0
00 => 00 => [2] => ([],2)
=> ? = 0
01 => 10 => [1,1] => ([(0,1)],2)
=> 1
10 => 01 => [1,1] => ([(0,1)],2)
=> 1
11 => 11 => [2] => ([],2)
=> ? = 0
000 => 000 => [3] => ([],3)
=> ? = 0
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
010 => 100 => [1,2] => ([(1,2)],3)
=> ? = 1
011 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
101 => 011 => [1,2] => ([(1,2)],3)
=> ? = 1
110 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
111 => 111 => [3] => ([],3)
=> ? = 0
0000 => 0000 => [4] => ([],4)
=> ? = 1
0001 => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0111 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
1010 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1110 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1111 => 1111 => [4] => ([],4)
=> ? = 1
00000 => 00000 => [5] => ([],5)
=> ? = 1
11111 => 11111 => [5] => ([],5)
=> ? = 1
000000 => 000000 => [6] => ([],6)
=> ? = 1
111111 => 111111 => [6] => ([],6)
=> ? = 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 117 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001432The order dimension of the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. 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. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001569The maximal modular displacement of a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. 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. 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 $eA$. St001545The second Elser number of a connected graph. St000806The semiperimeter of the associated bargraph. St001330The hat guessing number of a graph. St000699The toughness times the least common multiple of 1,. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000454The largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001060The distinguishing index of a graph. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
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