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Your data matches 549 different statistics following compositions of up to 3 maps.
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Matching statistic: St000237
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,4,1,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,4,6,1,3,5] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,6,2,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,6,1,5] => 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,6,2,5,4,1] => 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,6,5,3,1] => 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [5,4,2,1,6,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,3,1,6,2,4] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [5,6,2,1,3,4] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [6,5,2,4,1,3] => 0
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [6,3,5,1,4,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,6,1,5,3,2] => 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [5,1,6,4,3,2] => 0
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001159
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,1,0,0]
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
Description
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St000153
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [6,4,2,1,3,5] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,3,2,1,6,5] => [6,4,3,2,1,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => [6,4,3,2,1,5] => 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,3,5,6,4,2] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => [6,5,4,2,1,3] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => [6,5,4,2,1,3] => 0
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000212
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000212: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000212: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> []
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> []
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> []
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [3]
=> 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> 1
Description
The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row.
Summing over all partitions of $n$ yields the sequence
$$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$
which is [[oeis:A237770]].
The references in this sequence of the OEIS indicate a connection with Baxter permutations.
Matching statistic: St000234
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => [1,2] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,4,1,3] => [1,4,2,3] => 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,4,3,6,5,2] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,4,1,3,6,5] => [1,4,2,6,5,3] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => [3,2,1,6,5,4] => 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [4,3,2,6,1,5] => [3,2,1,6,4,5] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,2,1,5,4,3] => [2,6,5,4,1,3] => 0
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [6,2,5,1,4,3] => [2,6,5,3,1,4] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,3,6,2,1,5] => [4,2,1,6,3,5] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [1,5,2,6,3,4] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,4,1,6,3,5] => [1,5,2,6,4,3] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,4,6,3,5] => [1,5,3,6,4,2] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,6,5,4,3,2] => 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [2,6,1,5,4,3] => [1,6,5,4,2,3] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [2,6,5,1,4,3] => [1,6,5,3,2,4] => 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [2,6,5,4,1,3] => [1,6,4,3,2,5] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [5,4,3,2,1,6] => 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000274
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000274: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000274: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[(1,2),(3,4)]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[(1,4),(2,3)]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [[[[.,.],.],.],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [[[[.,.],.],[[.,.],.]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
Description
The number of perfect matchings of a graph.
A matching of a graph $G$ is a subset $F \subset E(G)$ such that no two edges in $F$ share a vertex in common. A perfect matching $F'$ is then a matching such that every vertex in $V(G)$ is incident with exactly one edge in $F'$.
Matching statistic: St000502
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,4,1,3] => {{1,2,4},{3}}
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => {{1,3,4},{2}}
=> 1
[(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,4,6,1,3,5] => {{1,2,4},{3,6},{5}}
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
=> 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,3,2,6,1,5] => {{1,4,6},{2,3},{5}}
=> 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => {{1,3,5,6},{2},{4}}
=> 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,6,2,5,4,1] => {{1,3},{2,6},{4,5}}
=> 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,6,5,3,1] => {{1,4,5},{2},{3,6}}
=> 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [5,4,2,1,6,3] => {{1,5,6},{2,4},{3}}
=> 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => {{1,4},{2},{3,5,6}}
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,3,1,6,2,4] => {{1,5},{2,3},{4,6}}
=> 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [5,6,2,1,3,4] => {{1,5},{2,6},{3},{4}}
=> 0
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [6,5,2,4,1,3] => {{1,6},{2,5},{3},{4}}
=> 0
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [6,3,5,1,4,2] => {{1,6},{2,3,5},{4}}
=> 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,6,1,5,3,2] => {{1,4,5},{2,6},{3}}
=> 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [5,1,6,4,3,2] => {{1,5},{2},{3,6},{4}}
=> 0
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}}
=> 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: St000546
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4,2,3,1] => [4,1,3,2] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [4,3,5,2,6,1] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,1,3,2,6,5] => [3,5,4,2,6,1] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,2,3,1,6,5] => [5,2,4,3,6,1] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [6,2,3,1,5,4] => [5,2,3,6,4,1] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,4,5,2,3,1] => [6,1,5,2,4,3] => 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,2,5,3,4,1] => [6,2,1,5,4,3] => 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [6,2,5,1,4,3] => [2,4,6,5,3,1] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [3,2,6,5,4,1] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [6,1,3,2,5,4] => [3,5,2,6,4,1] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => [4,3,2,6,5,1] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,4,5,3] => [4,3,6,1,5,2] => 0
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [6,4,5,1,3,2] => [2,6,5,1,4,3] => 0
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [6,1,5,3,4,2] => [3,6,1,5,4,2] => 0
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,3,5,1,4,2] => [2,6,3,5,4,1] => 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,3,5,2,4,1] => [6,1,3,5,4,2] => 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000989
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => [1,2] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [1,4,2,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => [1,4,2,3] => 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4,1,3,2] => [1,3,2,4] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,4,2,3,6,5] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => [1,4,2,6,3,5] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,1,3,2,6,5] => [1,3,2,6,4,5] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,1,3,2,6,4] => [1,3,2,6,4,5] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,1,3,2,5,4] => [1,3,2,5,4,6] => 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,1,4,2,5,3] => [1,4,2,5,3,6] => 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [5,1,4,2,6,3] => [1,4,2,6,3,5] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [1,5,2,6,3,4] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => [1,5,2,6,3,4] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => [1,5,2,3,6,4] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,3,5,4] => [1,6,2,3,5,4] => 0
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,1,6,2,5,4] => [1,6,2,5,3,4] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,1,6,2,5,3] => [1,6,2,5,3,4] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,2,5,1,4,3] => [1,4,2,5,3,6] => 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [5,2,6,1,4,3] => [1,4,2,6,3,5] => 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[(1,4),(2,3)]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 0
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
Description
Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
The following 539 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001357The maximal degree of a regular spanning subgraph of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000248The number of anti-singletons of a set partition. St000461The rix statistic of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000187The determinant of an alternating sign matrix. St000618The number of self-evacuating tableaux of given shape. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001271The competition number of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000065The number of entries equal to -1 in an alternating sign matrix. St000117The number of centered tunnels of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000879The number of long braid edges in the graph of braid moves of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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 $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000782The indicator function of whether a given perfect matching is an L & P matching. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001964The interval resolution global dimension of a poset. St000216The absolute length of a permutation. St000273The domination number of a graph. St000354The number of recoils of a permutation. St000472The sum of the ascent bottoms of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000544The cop number of a graph. St000653The last descent of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000916The packing number of a graph. St000917The open packing number of a graph. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001114The number of odd descents of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001672The restrained domination number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001948The number of augmented double ascents of a permutation. St000258The burning number of a graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000918The 2-limited packing number of a graph. St000990The first ascent of a permutation. St001330The hat guessing number of a graph. St001346The number of parking functions that give the same permutation. St001793The difference between the clique number and the chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001260The permanent of an alternating sign matrix. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001496The number of graphs with the same Laplacian spectrum as the given graph. 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. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000788The number of nesting-similar perfect matchings of a perfect matching. 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. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000787The number of flips required to make a perfect matching noncrossing. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. 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. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000628The balance of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000783The side length of the largest staircase partition fitting into a partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. 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. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001432The order dimension of the partition. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001484The number of singletons of an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001597The Frobenius rank of a skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001820The size of the image of the pop stack sorting operator. 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. 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. St000022The number of fixed points of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000402Half the size of the symmetry class of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001846The number of elements which do not have a complement in the lattice. St000929The constant term of the character polynomial of an integer partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000894The trace of an alternating sign matrix. St000880The number of connected components 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. St000068The number of minimal elements in a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000478Another weight of a partition according to Alladi. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000934The 2-degree of an integer partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000284The Plancherel distribution on integer partitions. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. 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. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St000958The number of Bruhat factorizations of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001890The maximum magnitude of the Möbius function of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000907The number of maximal antichains of minimal length in a poset. St000943The number of spots the most unlucky car had to go further in a parking function. St001429The number of negative entries in a signed permutation. St001520The number of strict 3-descents. St001555The order of a signed permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St000037The sign of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000633The size of the automorphism group of a poset. St000694The number of affine bounded permutations that project to a given permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. 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. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001399The distinguishing number of a poset. St001427The number of descents of a signed permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001851The number of Hecke atoms of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001864The number of excedances of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000102The charge of a semistandard tableau. St000221The number of strong fixed points of a permutation. St000239The number of small weak excedances. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000308The height of the tree associated to a permutation. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000962The 3-shifted major index of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001130The number of two successive successions in a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001430The number of positive entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001488The number of corners of a skew partition. St001524The degree of symmetry of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001545The second Elser number of a connected graph. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001684The reduced word complexity of a permutation. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001741The largest integer such that all patterns of this size are contained in the permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN 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. St001892The flag excedance statistic of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000401The size of the symmetry class of a permutation. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001817The number of flag weak exceedances of a signed permutation. 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 $eA$. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000100The number of linear extensions of a poset. St000137The Grundy value of an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000640The rank of the largest boolean interval in a poset. St000667The greatest common divisor of the parts of the 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000910The number of maximal chains of minimal length in a poset. St000944The 3-degree of an integer partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. 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. St001268The size of the largest ordinal summand in the poset. 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. St001360The number of covering relations in Young's lattice below a partition. 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. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001510The number of self-evacuating linear extensions of a finite poset. 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. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001541The Gini index of an integer 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. 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. 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. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001779The order of promotion on the set of linear extensions of a poset. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. St001961The sum of the greatest common divisors of all pairs of parts. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001409The maximal entry of a semistandard tableau. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001902The number of potential covers of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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