Your data matches 367 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000496
(load all 4 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => 0
[1,2] => [1,2] => {{1},{2}}
 => 0
[2,1] => [2,1] => {{1,2}}
 => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
 => 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
 => 0
[2,3,1] => [3,1,2] => {{1,2,3}}
 => 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
 => 1
[3,2,1] => [2,3,1] => {{1,2,3}}
 => 0
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
 => 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
 => 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
 => 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
 => 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
 => 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
 => 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
 => 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
 => 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
 => 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => 0
Description
The rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000752
(load all 5 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 0
[1,2] => [1,2] => [1,1]
 => 0
[2,1] => [2,1] => [2]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => 0
[2,3,1] => [3,1,2] => [2,1]
 => 0
[3,1,2] => [3,2,1] => [3]
 => 1
[3,2,1] => [2,3,1] => [2,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 0
[1,3,4,2] => [1,4,2,3] => [2,1,1]
 => 0
[1,4,2,3] => [1,4,3,2] => [3,1]
 => 1
[1,4,3,2] => [1,3,4,2] => [2,1,1]
 => 0
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 0
[2,1,4,3] => [2,1,4,3] => [2,2]
 => 0
[2,3,1,4] => [3,1,2,4] => [2,1,1]
 => 0
[2,3,4,1] => [4,1,2,3] => [2,1,1]
 => 0
[3,1,2,4] => [3,2,1,4] => [3,1]
 => 1
[3,2,1,4] => [2,3,1,4] => [2,1,1]
 => 0
[3,2,4,1] => [2,4,1,3] => [2,1,1]
 => 0
[3,4,1,2] => [3,1,4,2] => [2,2]
 => 0
[4,2,3,1] => [2,3,4,1] => [2,1,1]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1]
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => [2,1,1,1]
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => [2,2,1]
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => [2,2,1]
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => [2,2,1]
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => [2,1,1,1]
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => [3,1,1]
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => [2,1,1,1]
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => [2,2,1]
 => 0
Description
The Grundy value for the game 'Couples are forever' on an integer partition. Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St001172
(load all 7 compositions to match this statistic)
Mapping
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 0
[1,2] => [1,2] => [1,0,1,0]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 0
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 0
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 0
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 1
[1,4,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 0
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 0
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 0
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 0
[3,2,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,1,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 0
[4,2,3,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,4,3,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,4,5,2,3] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,5,3,4,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[2,1,5,4,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[3,1,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 1
[3,2,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[3,2,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 0
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001311
(load all 6 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
(load all 6 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
(load all 6 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001331
(load all 6 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001335
(load all 6 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001335: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
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: St001336
(load all 7 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St001736
(load all 6 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001736: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => 0
[2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
Description
The total number of cycles in a graph. For example, the complete graph on four vertices has four triangles and three different four-cycles.
The following 357 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001781The interlacing number of a set partition. St001797The number of overfull subgraphs of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000761The number of ascents in an integer composition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001512The minimum rank of a graph. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001845The number of join irreducibles minus the rank of a lattice. St000048The multinomial of the parts of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000346The number of coarsenings of a partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000883The number of longest increasing subsequences of a permutation. St000920The logarithmic height of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001261The Castelnuovo-Mumford regularity of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000290The major index of a binary word. St000291The number of descents of a binary word. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St000095The number of triangles of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001718The number of non-empty open intervals in a poset. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000299The number of nonisomorphic vertex-induced subtrees. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001933The largest multiplicity of a part in an integer partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St001394The genus of a permutation. St000223The number of nestings in the permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000732The number of double deficiencies of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000886The number of permutations with the same antidiagonal sums. St000779The tier of a permutation. St001727The number of invisible inversions of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000934The 2-degree of an integer partition. 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. St001175The size of a partition minus the hook length of the base cell. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000355The number of occurrences of the pattern 21-3. St001396Number of triples of incomparable elements in a finite poset. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000264The girth of a graph, which is not a tree. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000359The number of occurrences of the pattern 23-1. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000534The number of 2-rises of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. 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. St000284The Plancherel distribution on integer partitions. 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. St000735The last entry on the main diagonal of a standard tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000365The number of double ascents of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000317The cycle descent number of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000357The number of occurrences of the pattern 12-3. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000948The chromatic discriminant of a graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000806The semiperimeter of the associated bargraph. St000219The number of occurrences of the pattern 231 in a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001875The number of simple modules with projective dimension at most 1. St001964The interval resolution global dimension of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. 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. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000618The number of self-evacuating tableaux of given shape. 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. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements 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. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000456The monochromatic index of a connected graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001871The number of triconnected components of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001867The number of alignments of type EN of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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 $eA$.