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Your data matches 943 different statistics following compositions of up to 3 maps.
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Matching statistic: St000782
St000782: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> 1
[(1,4),(2,3),(5,6)]
=> 1
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 0
[(1,5),(2,4),(3,6)]
=> 1
[(1,4),(2,5),(3,6)]
=> 0
[(1,3),(2,5),(4,6)]
=> 0
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 1
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 1
[(1,5),(2,6),(3,4)]
=> 1
[(1,6),(2,5),(3,4)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001040
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001040: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> 3 = 0 + 3
[(1,3),(2,4),(5,6)]
=> 3 = 0 + 3
[(1,4),(2,3),(5,6)]
=> 3 = 0 + 3
[(1,5),(2,3),(4,6)]
=> 4 = 1 + 3
[(1,6),(2,3),(4,5)]
=> 4 = 1 + 3
[(1,6),(2,4),(3,5)]
=> 4 = 1 + 3
[(1,5),(2,4),(3,6)]
=> 4 = 1 + 3
[(1,4),(2,5),(3,6)]
=> 4 = 1 + 3
[(1,3),(2,5),(4,6)]
=> 4 = 1 + 3
[(1,2),(3,5),(4,6)]
=> 4 = 1 + 3
[(1,2),(3,6),(4,5)]
=> 4 = 1 + 3
[(1,3),(2,6),(4,5)]
=> 4 = 1 + 3
[(1,4),(2,6),(3,5)]
=> 4 = 1 + 3
[(1,5),(2,6),(3,4)]
=> 4 = 1 + 3
[(1,6),(2,5),(3,4)]
=> 4 = 1 + 3
Description
The depth of the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001195
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 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
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> 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 global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St000337
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3 = 1 + 2
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => 3 = 1 + 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 3 = 1 + 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => 3 = 1 + 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2 = 0 + 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => 3 = 1 + 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => 2 = 0 + 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => 3 = 1 + 2
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => 3 = 1 + 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => 3 = 1 + 2
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 3 = 1 + 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => 3 = 1 + 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => 2 = 0 + 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => 3 = 1 + 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 3 = 1 + 2
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000727
(load all 71 compositions to match this statistic)
(load all 71 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 5 = 0 + 5
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => 5 = 0 + 5
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 5 = 0 + 5
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => 6 = 1 + 5
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 6 = 1 + 5
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => 6 = 1 + 5
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => 6 = 1 + 5
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => 6 = 1 + 5
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => 6 = 1 + 5
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => 6 = 1 + 5
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 6 = 1 + 5
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => 6 = 1 + 5
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => 6 = 1 + 5
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => 6 = 1 + 5
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 6 = 1 + 5
Description
The largest label of a leaf in the binary search tree associated with the permutation.
Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
Matching statistic: St000205
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,2]
=> 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,2]
=> 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [4,2]
=> 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6]
=> 0
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6]
=> 0
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [4,2]
=> 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,2,2]
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,2]
=> 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,2]
=> 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [4,2]
=> 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [4,2]
=> 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,2]
=> 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6]
=> 0
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,2]
=> 1
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000237
(load all 4 compositions to match this statistic)
(load all 4 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),(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: St000247
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
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%
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),(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: St000352
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000461
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,1,3,2,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,1,3,2,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,1,3,2,5,4] => 0
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,1,4,2,5,3] => 0
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [5,1,4,2,6,3] => 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] => [3,1,5,2,6,4] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,3,5,4] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,1,6,2,5,4] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,1,6,2,5,3] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,2,5,1,4,3] => 0
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [5,2,6,1,4,3] => 1
Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
The following 933 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 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. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001549The number of restricted non-inversions between exceedances. St001557The number of inversions of the second entry of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000007The number of saliances of the permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000060The greater neighbor of the maximum. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000183The side length of the Durfee square of an integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000314The number of left-to-right-maxima of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000630The length of the shortest palindromic decomposition of a binary word. St000654The first descent of a permutation. St000701The protection number of a binary tree. St000710The number of big deficiencies of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000862The number of parts of the shifted shape of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. 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. St001469The holeyness of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001924The number of cells in an integer partition whose arm and leg length coincide. St001928The number of non-overlapping descents in a permutation. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000662The staircase size of the code of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000925The number of topologically connected components of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St000141The maximum drop size of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000702The number of weak deficiencies of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001670The connected partition number of a graph. St000019The cardinality of the support of a permutation. St000653The last descent of a permutation. St000735The last entry on the main diagonal of a standard tableau. St001807The lower middle entry of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000501The size of the first part in the decomposition of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000625The sum of the minimal distances to a greater element. St000471The sum of the ascent tops of a permutation. St001519The pinnacle sum of a permutation. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000021The number of descents of a permutation. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000055The inversion sum of a permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000133The "bounce" of a permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000153The number of adjacent cycles of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000159The number of distinct parts of the integer partition. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000216The absolute length of a permutation. St000221The number of strong fixed points of a permutation. St000222The number of alignments in the permutation. St000224The sorting index of a permutation. St000232The number of crossings of a set partition. St000234The number of global ascents of a permutation. St000238The number of indices that are not small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000248The number of anti-singletons of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000254The nesting number of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000274The number of perfect matchings of a graph. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000338The number of pixed points of a permutation. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000341The non-inversion sum of a permutation. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000409The number of pitchforks in a binary tree. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000446The disorder of a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000502The number of successions of a set partitions. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000534The number of 2-rises of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000546The number of global descents of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000618The number of self-evacuating tableaux of given shape. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000650The number of 3-rises of a permutation. St000651The maximal size of a rise in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000672The number of minimal elements in Bruhat order not less than the permutation. St000674The number of hills of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St000682The Grundy value of Welter's game on a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. 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. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in 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. St000834The number of right outer peaks of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001153The number of blocks with even minimum in a set partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $K[x]/(x^n)$. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for 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. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. 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. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. St001375The pancake length of a permutation. St001388The number of non-attacking neighbors of a permutation. St001394The genus of a permutation. St001403The number of vertical separators in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001423The number of distinct cubes in a binary word. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 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. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001513The number of nested exceedences of a permutation. St001524The degree of symmetry of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001556The number of inversions of the third entry of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort 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. 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. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001669The number of single rises in a Dyck path. St001671Haglund's hag of a permutation. St001673The degree of asymmetry of an integer composition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001684The reduced word complexity of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001715The number of non-records in a permutation. St001726The number of visible inversions of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001850The number of Hecke atoms of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001911A descent variant minus the number of inversions. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. 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. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000020The rank of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000040The number of regions of the inversion arrangement of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000079The number of alternating sign matrices for a given Dyck path. St000080The rank of the poset. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000257The number of distinct parts of a partition that occur at least twice. St000260The radius of a connected graph. St000273The domination number of a graph. St000277The number of ribbon shaped standard tableaux. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000335The difference of lower and upper interactions. St000351The determinant of the adjacency matrix of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000382The first part of an integer composition. St000390The number of runs of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000402Half the size of the symmetry class of a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000484The sum of St000483 over all subsequences of length at least three. 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. St000526The number of posets with combinatorially isomorphic order polytopes. St000530The number of permutations with the same descent word as the given permutation. St000531The leading coefficient of the rook polynomial of an integer partition. St000545The number of parabolic double cosets with minimal element being the given permutation. St000619The number of cyclic descents of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000628The balance of a binary word. St000638The number of up-down runs of a permutation. St000640The rank of the largest boolean interval in a poset. St000668The least common multiple of the parts of the partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000700The protection number of an ordered tree. St000711The number of big exceedences of a permutation. St000740The last entry of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000820The number of compositions obtained by rotating the composition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000906The length of the shortest maximal chain in a poset. St000913The number of ways to refine the partition into singletons. St000919The number of maximal left branches of a binary tree. St000920The logarithmic height of a Dyck path. St000928The sum of the coefficients of the character polynomial of an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000958The number of Bruhat factorizations of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001052The length of the exterior of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001160The number of proper blocks (or intervals) of a permutations. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001203We associate to 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 Dyck path as follows:
St001220The width of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. St001280The number of parts of an integer partition that are at least two. St001285The number of primes in the column sums of the two line notation of a permutation. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001346The number of parking functions that give the same permutation. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001405The number of bonds in a permutation. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001471The magnitude of a Dyck path. St001482The product of the prefix sums of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001652The length of a longest interval of consecutive numbers. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001716The 1-improper chromatic number of a graph. St001727The number of invisible inversions of a permutation. St001732The number of peaks visible from the left. St001741The largest integer such that all patterns of this size are contained in the permutation. St001777The number of weak descents in an integer composition. St001778The largest greatest common divisor of an element and its image in a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001792The arboricity of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001806The upper middle entry of a permutation. St001829The common independence number of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001884The number of borders of a binary word. 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. St001958The degree of the polynomial interpolating the values of a permutation. St000024The number of double up and double down steps of a Dyck path. St000050The depth or height of a binary tree. St000064The number of one-box pattern of a permutation. St000075The orbit size of a standard tableau under promotion. St000093The cardinality of a maximal independent set of vertices of a graph. St000203The number of external nodes of a binary tree. St000211The rank of the set partition. St000236The number of cyclical small weak excedances. St000362The size of a minimal vertex cover of a graph. St000381The largest part of an integer composition. St000387The matching number of a graph. St000392The length of the longest run of ones in a binary word. St000398The sum of the depths of the vertices (or total internal path length) of a binary tree. St000491The number of inversions of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000565The major index of a set partition. St000568The hook number of a binary tree. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000734The last entry in the first row of a standard tableau. St000737The last entry on the main diagonal of a semistandard tableau. St000778The metric dimension of a graph. St000780The size of the orbit under rotation of a perfect matching. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000918The 2-limited packing number of a graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000945The number of matchings in the dihedral orbit of a perfect matching. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000969We 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 $c_{n-1}$. St000971The smallest closer of a set partition. St000981The length of the longest zigzag subpath. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001050The number of terminal closers of a set partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001214The aft of an integer partition. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001402The number of separators in a permutation. St001486The number of corners of the ribbon associated with an integer composition. St001500The global dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001516The number of cyclic bonds of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001527The cyclic permutation representation number of an integer partition. St001554The number of distinct nonempty subtrees of a binary tree. St001566The length of the longest arithmetic progression in a permutation. St001642The Prague dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001734The lettericity of a graph. St001812The biclique partition number of a graph. St001910The height of the middle non-run of a Dyck path. St001931The weak major index of an integer composition regarded as a word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000051The size of the left subtree of a binary tree. St000105The number of blocks in the set partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000170The trace of a semistandard tableau. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000384The maximal part of the shifted composition of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000503The maximal difference between two elements in a common block. St000519The largest length of a factor maximising the subword complexity. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000652The maximal difference between successive positions of a permutation. St000676The number of odd rises of a Dyck path. St000730The maximal arc length of a set partition. St000784The maximum of the length and the largest part of the integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001007Number of simple modules with projective dimension 1 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. St001083The number of boxed occurrences of 132 in a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001391The disjunction number of a graph. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001672The restrained domination number of a graph. St000013The height of a Dyck path. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000342The cosine of a permutation. St000345The number of refinements of a partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000438The position of the last up step in a Dyck path. St000459The hook length of the base cell of a partition. St000520The number of patterns in a permutation. St000626The minimal period of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000922The minimal number such that all substrings of this length are unique. St000937The number of positive values of the symmetric group character corresponding to the partition. St000982The length of the longest constant subword. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001389The number of partitions of the same length below the given integer partition. St001411The number of patterns 321 or 3412 in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001480The number of simple summands of the module J^2/J^3. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St000026The position of the first return of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000806The semiperimeter of the associated bargraph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001268The size of the largest ordinal summand in the poset. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001468The smallest fixpoint of a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000493The los statistic of a set partition. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000867The sum of the hook lengths in the first row of an integer partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St000747A variant of the major index of a set partition. St000399The external path length of a binary tree. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St001293The sum of all $1/(i+\pi(i))$ for a permutation $\pi$ times the lcm of all possible values among permutations of the same length. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000152The number of boxed plus the number of special entries. St000231Sum of the maximal elements of the blocks of a set partition. St000869The sum of the hook lengths of an integer partition. St000294The number of distinct factors of a binary word. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001271The competition number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000370The genus of a graph. St001176The size of a partition minus its first part. St001657The number of twos in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000187The determinant of an alternating sign matrix. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. 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 $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001128The exponens consonantiae of a partition. 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 $g$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice. 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. St000544The cop number of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001638The book thickness of a graph. 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. St000088The row sums of the character table of the symmetric group. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. 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. St000346The number of coarsenings of a partition. St000454The largest eigenvalue of a graph if it is integral. St000517The Kreweras number of an integer partition. St000549The number of odd partial sums of an integer partition. 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. St000741The Colin de Verdière graph invariant. 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. St000929The constant term of the character polynomial of an integer partition. 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. 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. St001129The product of the squares of the parts of a partition. 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. St001260The permanent of an alternating sign matrix. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001368The number of vertices of maximal degree in a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001481The minimal height of a peak of a Dyck path. 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. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. 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. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000455The second largest eigenvalue of a graph if it is integral. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St000880The number of connected components 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. 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. St000934The 2-degree of an integer partition. St000284The Plancherel distribution on integer partitions. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000567The sum of the products of all pairs of parts. 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. 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. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. 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. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001615The number of join prime elements of a lattice. St001964The interval resolution global dimension of a poset. St000894The trace of an alternating sign matrix. St001846The number of elements which do not have a complement in the lattice. 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. 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. St000225Difference between largest and smallest parts in a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001279The sum of the 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. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. 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. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001763The Hurwitz number of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001943The sum of the squares of the hook lengths of an integer partition.
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