Your data matches 211 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000247
(load all 8 compositions to match this statistic)
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000561
(load all 8 compositions to match this statistic)
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => 1
Description
The number of occurrences of the pattern {{1,2,3}} in a set partition.
Matching statistic: St001062
(load all 8 compositions to match this statistic)
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => 2 = 0 + 2
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 3 = 1 + 2
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 2 = 0 + 2
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => 3 = 1 + 2
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => 3 = 1 + 2
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => 3 = 1 + 2
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => 3 = 1 + 2
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => 2 = 0 + 2
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => 3 = 1 + 2
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => 3 = 1 + 2
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => 3 = 1 + 2
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => 2 = 0 + 2
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => 3 = 1 + 2
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => 3 = 1 + 2
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => 3 = 1 + 2
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => 2 = 0 + 2
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => 3 = 1 + 2
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => 3 = 1 + 2
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => 3 = 1 + 2
Description
The maximal size of a block of a set partition.
Matching statistic: St000022
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => [2,1] => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => [5,4,3,2,6,1] => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => [5,3,2,6,1,4] => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => [5,2,4,6,1,3] => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => [4,2,6,5,1,3] => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => [4,2,5,1,6,3] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000248
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => {{1},{2}}
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => {{1,4},{2,3,5},{6}}
 => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => {{1,4},{2,3,6},{5}}
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => {{1,5},{2,3,6},{4}}
 => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,4,6},{3}}
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => {{1,5},{2},{3,4,6}}
 => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => {{1},{2},{3,6},{4,5}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => {{1},{2,6},{3},{4,5}}
 => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => {{1},{2,6},{3,5},{4}}
 => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => {{1},{2,6},{3,4},{5}}
 => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => {{1},{2,5},{3,4},{6}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => {{1,3},{2,4,5},{6}}
 => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,4,6},{5}}
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => {{1,3},{2,5,6},{4}}
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => {{1,4},{2,5,6},{3}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => {{1,4},{2},{3,5,6}}
 => 1
Description
The number of anti-singletons of a set partition. An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$. For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000475
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => [2]
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1]
 => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2]
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => [3,1]
 => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => [3,2,1]
 => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => [3,2,1]
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => [3,2,1]
 => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => [2,2,2]
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => [3,2,1]
 => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => [3,2,1]
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => [3,2,1]
 => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => [2,2,2]
 => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => [3,2,1]
 => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => [3,2,1]
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => [3,2,1]
 => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => [2,2,2]
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => [3,2,1]
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => [3,2,1]
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => [3,2,1]
 => 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000486
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => [2,1] => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => [5,4,3,2,6,1] => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => [5,3,2,6,1,4] => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => [5,2,4,6,1,3] => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => [4,2,6,5,1,3] => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => [4,2,5,1,6,3] => 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000502
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => {{1},{2}}
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => {{1,4},{2,3,5},{6}}
 => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => {{1,4},{2,3,6},{5}}
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => {{1,5},{2,3,6},{4}}
 => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,4,6},{3}}
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => {{1,5},{2},{3,4,6}}
 => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => {{1},{2},{3,6},{4,5}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => {{1},{2,6},{3},{4,5}}
 => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => {{1},{2,6},{3,5},{4}}
 => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => {{1},{2,6},{3,4},{5}}
 => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => {{1},{2,5},{3,4},{6}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => {{1,3},{2,4,5},{6}}
 => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,4,6},{5}}
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => {{1,3},{2,5,6},{4}}
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => {{1,4},{2,5,6},{3}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => {{1,4},{2},{3,5,6}}
 => 1
Description
The number of successions of a set partitions. This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000731
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => [2,1] => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => [5,4,3,2,6,1] => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => [6,3,5,4,2,1] => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => [5,3,2,6,1,4] => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => [5,2,4,6,1,3] => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => [4,2,6,5,1,3] => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => [4,2,5,1,6,3] => 1
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000752
Mapping
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => {{1,2}}
 => {{1,2}}
 => [2]
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1]
 => 1
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [2,2]
 => 0
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => [3,1]
 => 1
[(1,2),(3,4),(5,6)]
 => {{1,2},{3,4},{5,6}}
 => {{1,2,4},{3,6},{5}}
 => [3,2,1]
 => 1
[(1,3),(2,4),(5,6)]
 => {{1,3},{2,4},{5,6}}
 => {{1,4},{2,3,6},{5}}
 => [3,2,1]
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3,4},{2,6},{5}}
 => [3,2,1]
 => 1
[(1,5),(2,3),(4,6)]
 => {{1,5},{2,3},{4,6}}
 => {{1,3},{2,6},{4,5}}
 => [2,2,2]
 => 0
[(1,6),(2,3),(4,5)]
 => {{1,6},{2,3},{4,5}}
 => {{1,3},{2,5,6},{4}}
 => [3,2,1]
 => 1
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5,6},{2,4},{3}}
 => [3,2,1]
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4,5},{3}}
 => [3,2,1]
 => 1
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => [2,2,2]
 => 0
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3,5},{4}}
 => [3,2,1]
 => 1
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2,6},{3,5},{4}}
 => [3,2,1]
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2,5},{3},{4,6}}
 => [3,2,1]
 => 1
[(1,3),(2,6),(4,5)]
 => {{1,3},{2,6},{4,5}}
 => {{1,5},{2,3},{4,6}}
 => [2,2,2]
 => 0
[(1,4),(2,6),(3,5)]
 => {{1,4},{2,6},{3,5}}
 => {{1,5},{2},{3,4,6}}
 => [3,2,1]
 => 1
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4,5},{2},{3,6}}
 => [3,2,1]
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2},{3,5,6}}
 => [3,2,1]
 => 1
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.
The following 201 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001673The degree of asymmetry of an integer composition. St001810The number of fixed points of a permutation smaller than its largest moved point. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000147The largest part of an integer partition. St000381The largest part of an integer composition. St000485The length of the longest cycle of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000260The radius of a connected graph. 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 $eA$. St001330The hat guessing number of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. 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. St000744The length of the path to the largest entry in a standard Young tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000298The order dimension or Dushnik-Miller dimension of a poset. 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. St000640The rank of the largest boolean interval in a poset. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000931The number of occurrences of the pattern UUU in 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. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001545The second Elser number of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001557The number of inversions of the second entry of a permutation. St001822The number of alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St000455The second largest eigenvalue of a graph if it is integral. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St001060The distinguishing index of a graph. St001152The number of pairs with even minimum in a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001427The number of descents of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St000893The number of distinct diagonal sums of an alternating sign matrix. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001812The biclique partition number of a graph. St001875The number of simple modules with projective dimension at most 1. 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. 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. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. 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. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. St001964The interval resolution global dimension of a poset. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St001618The cardinality of the Frattini sublattice of a lattice. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.