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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001645
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: 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
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
1 => 1 => [1,1] => ([(0,1)],2)
=> 2
00 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 4
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
0000 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
1111 => 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)
=> 5
00000 => 01111 => [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)
=> 6
00001 => 10111 => [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)
=> 6
11110 => 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)
=> 6
11111 => 11111 => [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)
=> 6
000000 => 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
000001 => 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
000010 => 000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
000011 => 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
111100 => 010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
111101 => 100111 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
111110 => 001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
111111 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
=> => [1] => ([],1)
=> 1
Description
The pebbling number of a connected graph.
Matching statistic: St001880
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001880: Posets ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Mp00262: Binary words —poset of factors⟶ Posets
St001880: Posets ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Values
1 => 0 => ([(0,1)],2)
=> ? = 2
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => 00 => ([(0,2),(2,1)],3)
=> 3
111 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
1111 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 6
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
11111 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 7
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 7
000011 => 000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 7
111100 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111101 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
111111 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
=> => ?
=> ? = 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001914
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 42%●distinct values known / distinct values provided: 29%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 42%●distinct values known / distinct values provided: 29%
Values
1 => [1] => [1,0]
=> []
=> ? = 2 - 2
00 => [2] => [1,1,0,0]
=> []
=> ? = 4 - 2
11 => [2] => [1,1,0,0]
=> []
=> ? = 3 - 2
111 => [3] => [1,1,1,0,0,0]
=> []
=> ? = 4 - 2
0000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 5 - 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 5 - 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 6 - 2
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 6 - 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 6 - 2
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 6 - 2
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 7 - 2
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 5 = 7 - 2
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 5 = 7 - 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 5 = 7 - 2
111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 5 = 7 - 2
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 5 = 7 - 2
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 5 = 7 - 2
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 7 - 2
=> [] => ?
=> ?
=> ? = 1 - 2
Description
The size of the orbit of an integer partition in Bulgarian solitaire.
Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
This statistic returns the number of partitions that can be obtained from the given partition.
Matching statistic: St000189
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000189: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Mp00262: Binary words —poset of factors⟶ Posets
St000189: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Values
1 => 0 => ([(0,1)],2)
=> 2
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => 00 => ([(0,2),(2,1)],3)
=> 3
111 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
1111 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 6
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
11111 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 7
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 7
000011 => 000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 7
111100 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111101 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
111111 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
=> => ?
=> ? = 1
Description
The number of elements in the poset.
Matching statistic: St000656
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000656: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Mp00262: Binary words —poset of factors⟶ Posets
St000656: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Values
1 => 0 => ([(0,1)],2)
=> 2
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => 00 => ([(0,2),(2,1)],3)
=> 3
111 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
1111 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 6
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
11111 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 7
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 7
000011 => 000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 7
111100 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111101 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
111111 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
=> => ?
=> ? = 1
Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Matching statistic: St001717
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001717: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Mp00262: Binary words —poset of factors⟶ Posets
St001717: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Values
1 => 0 => ([(0,1)],2)
=> 2
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => 00 => ([(0,2),(2,1)],3)
=> 3
111 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
1111 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 6
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
11111 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 7
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 7
000011 => 000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 7
111100 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111101 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
111111 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
=> => ?
=> ? = 1
Description
The largest size of an interval in a poset.
Matching statistic: St001300
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001300: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Mp00262: Binary words —poset of factors⟶ Posets
St001300: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 71%
Values
1 => 0 => ([(0,1)],2)
=> 1 = 2 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
11 => 00 => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
111 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
0000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 - 1
1111 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
00000 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
00001 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 6 - 1
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
11111 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
000000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7 - 1
000001 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 7 - 1
000010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 7 - 1
000011 => 000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 7 - 1
111100 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7 - 1
111101 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 7 - 1
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 - 1
111111 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 - 1
=> => ?
=> ? = 1 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000777
Values
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 2
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 4
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 3
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
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)
=> 5
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)
=> 5
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)
=> 6
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 6
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ([(0,1),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 6
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)
=> 6
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)
=> 7
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,10),(1,11),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(3,4),(3,5),(3,7),(3,8),(3,9),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 7
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ([(0,3),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,10),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 7
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(0,7),(0,11),(1,6),(1,10),(2,8),(2,10),(3,9),(3,11),(4,8),(4,9),(4,14),(5,6),(5,7),(5,14),(6,12),(7,13),(8,12),(9,13),(10,12),(11,13),(12,14),(13,14)],15)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,14),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 7
111100 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(0,7),(0,11),(1,6),(1,10),(2,8),(2,10),(3,9),(3,11),(4,8),(4,9),(4,14),(5,6),(5,7),(5,14),(6,12),(7,13),(8,12),(9,13),(10,12),(11,13),(12,14),(13,14)],15)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,14),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 7
111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ([(0,3),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,4),(2,6),(2,7),(2,8),(2,9),(2,10),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 7
111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,10),(1,11),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(3,4),(3,5),(3,7),(3,8),(3,9),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,8),(5,9),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 7
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)
=> 7
=> ?
=> ?
=> ?
=> ? = 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
1 => 1 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
00 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 4 - 1
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
0000 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 1
1111 => 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)
=> 4 = 5 - 1
00000 => 01111 => [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)
=> ? = 6 - 1
00001 => 10111 => [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)
=> ? = 6 - 1
11110 => 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)
=> ? = 6 - 1
11111 => 11111 => [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)
=> 5 = 6 - 1
000000 => 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
000001 => 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
000010 => 000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
000011 => 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
111100 => 010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
111101 => 100111 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
111110 => 001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 - 1
111111 => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 7 - 1
=> => [1] => ([],1)
=> 0 = 1 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001879
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Values
1 => [1] => [[1],[]]
=> ([],1)
=> ? = 2 - 2
00 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 4 - 2
11 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 3 - 2
111 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
0000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
1111 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
00000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
00001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
11110 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 2
11111 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
000000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
000001 => [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 7 - 2
000010 => [4,1,1] => [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 7 - 2
000011 => [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ? = 7 - 2
111100 => [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ? = 7 - 2
111101 => [4,1,1] => [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 7 - 2
111110 => [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 7 - 2
111111 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 7 - 2
=> [] => ?
=> ?
=> ? = 1 - 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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 $eA$.
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