Your data matches 895 different statistics following compositions of up to 3 maps.
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Mp00201: Dyck paths RingelPermutations
Mp00160: Permutations graph of inversionsGraphs
St000454: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[]
=> [1] => ([],1)
=> 0
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00159: Permutations Demazure product with inversePermutations
St000891: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,4,3,2,1] => 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [5,4,3,2,1,6] => 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 2
[]
=> [] => [] => 0
Description
The number of distinct diagonal sums of a permutation matrix. For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$$ are $(1,0,1,0,2,0)$, so the statistic is $3$.
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [1,2,5,3,4] => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2,3,6,1,5] => [2,3,1,6,4,5] => 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [2,4,3,1,6,5] => [1,3,2,6,5,4] => 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,5,6,4] => 2
[]
=> [] => [] => [] => 0
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
Mp00142: Dyck paths promotionDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 2
[]
=> []
=> []
=> []
=> 0
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000147
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00108: Permutations cycle typeInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [2,2]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,6,5,4,3,2] => [2,2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [2,6,5,4,3,1] => [3,2,1]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => [3,2,1,6,5,4] => [2,2,1,1]
=> 2
[]
=> [] => [] => []
=> 0
Description
The largest part of an integer partition.
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00088: Permutations Kreweras complementPermutations
St000153: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [1,2,4,3] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [2,1,5,4,3] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => [4,3,2,1,5] => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,6,5,4,3,2] => [2,1,6,5,4,3] => 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [2,6,5,4,3,1] => [1,2,6,5,4,3] => 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => [3,2,1,6,5,4] => [4,3,2,1,6,5] => 2
[]
=> [] => [] => [] => 0
Description
The number of adjacent cycles of a permutation. This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Mp00201: Dyck paths RingelPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00160: Permutations graph of inversionsGraphs
St000259: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,1,4,2,5,3] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [7,1,4,2,3,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [6,3,4,1,2,7,5] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [7,2,3,5,1,6,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[]
=> [1] => [1] => ([],1)
=> 0
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000507
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [[1,3],[2,4],[5]]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => [[1,3],[2,4],[5],[6]]
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => [[1,2],[3,4],[5],[6]]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 2
[]
=> [] => [] => []
=> 0
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00160: Permutations graph of inversionsGraphs
St001270: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
[]
=> []
=> [1] => ([],1)
=> 0
Description
The bandwidth of a graph. The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$. We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Mp00201: Dyck paths RingelPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001502: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[]
=> [1] => [1]
=> [1,0,1,0]
=> 0
Description
The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. We use the code below to translate them to Dyck paths. The algebras where the statistic returns 0 are exactly the higher Auslander algebras and are of special interest. It seems like they are counted by the number of divisors function.
The following 885 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000005The bounce statistic of a Dyck path. St000947The major index east count of a Dyck path. St001108The 2-dynamic chromatic number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St000236The number of cyclical small weak excedances. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001119The length of a shortest maximal path in a graph. St001128The exponens consonantiae of a partition. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000365The number of double ascents of a permutation. St000651The maximal size of a rise in a permutation. St000834The number of right outer peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. 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 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000013The height of a Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000239The number of small weak excedances. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000638The number of up-down runs of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000808The number of up steps of the associated bargraph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama 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: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001261The Castelnuovo-Mumford regularity of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001642The Prague dimension of a graph. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001806The upper middle entry of a permutation. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000069The number of maximal elements of a poset. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000155The number of exceedances (also excedences) of a permutation. St000160The multiplicity of the smallest part of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000439The position of the first down step of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000565The major index of a set partition. St000628The balance of a binary word. St000647The number of big descents of a permutation. St000662The staircase size of the code of a permutation. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000769The major index of a composition regarded as a word. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000920The logarithmic height of a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001096The size of the overlap set of a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. 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. St001489The maximum of the number of descents and the number of inverse descents. 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. St001565The number of arithmetic progressions of length 2 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001729The number of visible descents 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. St001801Half the number of preimage-image pairs of different parity in a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000886The number of permutations with the same antidiagonal sums. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000779The tier of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000872The number of very big descents of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St000485The length of the longest cycle of a permutation. 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 $eA$. St000124The cardinality of the preimage of the Simion-Schmidt map. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000402Half the size of the symmetry class of a permutation. St000461The rix statistic of a permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St000570The Edelman-Greene number of a permutation. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000919The number of maximal left branches of a binary tree. St000983The length of the longest alternating subword. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001052The length of the exterior of a permutation. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001566The length of the longest arithmetic progression in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000219The number of occurrences of the pattern 231 in a permutation. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000358The number of occurrences of the pattern 31-2. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000516The number of stretching pairs 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. St000562The number of internal points of a set partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000691The number of changes of a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in 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. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000956The maximal displacement of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001388The number of non-attacking neighbors of a permutation. St001394The genus of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000060The greater neighbor of the maximum. St000083The number of left oriented leafs of a binary tree except the first one. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000271The chromatic index of a graph. St000444The length of the maximal rise of a Dyck path. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000822The Hadwiger number of the graph. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St000842The breadth of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001062The maximal size of a block of a set partition. St001112The 3-weak dynamic number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001316The domatic number of a graph. St001346The number of parking functions that give the same permutation. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001655The general position number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001792The arboricity of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000040The number of regions of the inversion arrangement of a permutation. St000045The number of linear extensions of a binary tree. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. 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. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000310The minimal degree of a vertex of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000487The length of the shortest cycle of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000568The hook number of a binary tree. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000694The number of affine bounded permutations that project to a given permutation. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000729The minimal arc length of a set 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. St000778The metric dimension of a graph. St000798The makl of a permutation. St000820The number of compositions obtained by rotating the composition. St000823The number of unsplittable factors of the set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000833The comajor index of a permutation. St000837The number of ascents of distance 2 of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000903The number of different parts of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000933The number of multipartitions of sizes given by an integer partition. St000988The orbit size of a permutation under Foata's bijection. St000993The multiplicity of the largest part of an integer partition. 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. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001075The minimal size of a block of a set partition. St001081The number of minimal length factorizations of a permutation into star transpositions. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001114The number of odd descents of a permutation. St001162The minimum jump of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001277The degeneracy of a graph. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001313The number of Dyck paths above the lattice path given by a binary word. St001315The dissociation number of a graph. St001350Half of the Albertson index of a graph. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001469The holeyness of a permutation. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001577The minimal number of edges to add or remove to make a graph a cograph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. 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. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001959The product of the heights of the peaks of a Dyck path. St001962The proper pathwidth of a graph. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000055The inversion sum of a permutation. St000096The number of spanning trees of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St000221The number of strong fixed points of a permutation. St000224The sorting index of a permutation. St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000247The number of singleton blocks of a set partition. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000290The major index 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. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 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. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000478Another weight of a partition according to Alladi. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000503The maximal difference between two elements in a common block. St000538The number of even inversions of a permutation. St000552The number of cut vertices of a graph. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000572The dimension exponent of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. 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. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (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. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 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. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000730The maximal arc length of a set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000792The Grundy value for the game of ruler on a binary word. St000795The mad of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000809The reduced reflection length of the permutation. St000850The number of 1/2-balanced pairs in a poset. St000877The depth of the binary word interpreted as a path. 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. St000921The number of internal inversions of a binary word. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St000948The chromatic discriminant of a graph. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001082The number of boxed occurrences of 123 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001323The independence gap of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001377The major index minus the number of inversions of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001480The number of simple summands of the module J^2/J^3. St001485The modular major index of a binary word. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001525The number of symmetric hooks on the diagonal of a partition. St001557The number of inversions of the second entry of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001569The maximal modular displacement of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001726The number of visible inversions of a permutation. St001728The number of invisible 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. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001811The Castelnuovo-Mumford regularity of a permutation. St001850The number of Hecke atoms of a permutation. St001931The weak major index of an integer composition regarded as a word. 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. St000213The number of weak exceedances (also weak excedences) of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001933The largest multiplicity of a part in an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000120The number of left tunnels of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000741The Colin de Verdière graph invariant. St000893The number of distinct diagonal sums of an alternating sign matrix. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001118The acyclic chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St001405The number of bonds in a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000031The number of cycles in the cycle decomposition of a permutation. St000264The girth of a graph, which is not a tree. St000443The number of long tunnels of a Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000733The row containing the largest entry of a standard tableau. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000897The number of different multiplicities of parts of an integer partition. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001115The number of even descents of a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001439The number of even weak deficiencies and of odd weak exceedences. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of 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. St001571The Cartan determinant of the integer partition. St001732The number of peaks visible from the left. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000117The number of centered tunnels of a Dyck path. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000257The number of distinct parts of a partition that occur at least twice. St000379The number of Hamiltonian cycles in a graph. St000477The weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St001022Number of simple modules with projective dimension 3 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001561The value of the elementary symmetric function evaluated at 1. St001587Half of the largest even part of an integer partition. St001625The Möbius invariant of a lattice. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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. St000455The second largest eigenvalue of a graph if it is integral. St000849The number of 1/3-balanced pairs in a poset. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000015The number of peaks of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000746The number of pairs with odd minimum in a perfect matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001432The order dimension of the partition. St001488The number of corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St000023The number of inner peaks of a permutation. St000237The number of small exceedances. St000475The number of parts equal to 1 in a partition. St000731The number of double exceedences of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $g$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001516The number of cyclic bonds of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. 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). St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000225Difference between largest and smallest parts in a partition. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001862The number of crossings of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001402The number of separators in a permutation. St001427The number of descents of a signed permutation. St001487The number of inner corners of a skew partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001668The number of points of the poset minus the width of the poset. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000547The number of even non-empty partial sums of an integer partition. St000635The number of strictly order preserving maps of a poset into itself. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000913The number of ways to refine the partition into singletons. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. 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 $c_{n−1}$. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001435The number of missing boxes in the first row. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001597The Frobenius rank of a skew partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001780The order of promotion on the set of standard tableaux of given shape. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000017The number of inversions of a standard tableau. 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. St000567The sum of the products of all pairs of parts. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001098The 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 vertex labelled trees. 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. St001159Number of simple modules with dominant dimension equal to the global dimension 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 $K[x]/(x^n)$. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001248Sum of the even parts of a partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001596The number of two-by-two squares inside a skew partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000145The Dyson rank of a partition. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St000474Dyson's crank of a partition. St001415The length of the longest palindromic prefix of a binary word. St001890The maximum magnitude of the Möbius function of a poset. St000307The number of rowmotion orbits of a poset. St000381The largest part of an integer composition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000942The number of critical left to right maxima of the parking functions. St001462The number of factors of a standard tableaux under concatenation. St001545The second Elser number of a connected graph. St001637The number of (upper) dissectors of a poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001863The number of weak excedances of a signed permutation. St000100The number of linear extensions of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000632The jump number of the poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000871The number of very big ascents of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000981The length of the longest zigzag subpath. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. 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. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001267The length of the Lyndon factorization of the binary word. St001389The number of partitions of the same length below the given integer partition. St001490The number of connected components of a skew partition. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001946The number of descents in a parking function. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000848The balance constant multiplied with the number of linear extensions of a poset. St001249Sum of the odd parts of a partition. St001438The number of missing boxes of a skew 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. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000146The Andrews-Garvan crank of a partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000681The Grundy value of Chomp on Ferrers diagrams. St000717The number of ordinal summands of a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000680The Grundy value for Hackendot on posets. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000509The diagonal index (content) of a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000438The position of the last up step in a Dyck path. St000490The intertwining number of a set partition. St000679The pruning number of an ordered tree. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001624The breadth of a lattice. St001820The size of the image of the pop stack sorting operator. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000693The modular (standard) major index of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001500The global dimension of magnitude 1 Nakayama algebras. 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001763The Hurwitz number of an integer partition. St001808The box weight or horizontal decoration of a Dyck path. St001875The number of simple modules with projective dimension at most 1. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001943The sum of the squares of the hook lengths of an integer partition. St000327The number of cover relations in a poset. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000977MacMahon's equal index of a Dyck path. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001279The sum of the parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000656The number of cuts of a poset. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001854The size of the left Kazhdan-Lusztig cell, St001858The number of covering elements of a signed permutation in absolute order. St000639The number of relations in a poset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.