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Matching statistic: St001165
(load all 393 compositions to match this statistic)
(load all 393 compositions to match this statistic)
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001471
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(load all 392 compositions to match this statistic)
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St001011
(load all 624 compositions to match this statistic)
(load all 624 compositions to match this statistic)
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001188
(load all 264 compositions to match this statistic)
(load all 264 compositions to match this statistic)
St001188: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The 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.
Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.
Matching statistic: St001244
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(load all 264 compositions to match this statistic)
St001244: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.
For the projective dimension see [[St001007]] and for 1-regularity see [[St001126]]. After applying the inverse zeta map [[Mp00032]], this statistic matches the number of rises of length at least 2 [[St000659]].
Matching statistic: St001507
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(load all 393 compositions to match this statistic)
St001507: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.
Matching statistic: St000325
(load all 189 compositions to match this statistic)
(load all 189 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 2
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [2,1,3] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000397
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(load all 139 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000397: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1
[1,0,1,0]
=> [[],[]]
=> 2
[1,1,0,0]
=> [[[]]]
=> 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 2
[1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 1
Description
The Strahler number of a rooted tree.
Matching statistic: St000470
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(load all 189 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 2
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [2,1,3] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000686
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(load all 77 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
The finitistic dominant dimension of a Dyck path.
To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
The following 1177 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000862The number of parts of the shifted shape of a permutation. 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:
St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000306The bounce count of a Dyck path. St000711The number of big exceedences of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001394The genus of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000679The pruning number of an ordered tree. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000701The protection number of a binary tree. 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. St000822The Hadwiger number of the graph. St000920The logarithmic height of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001500The global dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001717The largest size of an interval in a poset. St001732The number of peaks visible from the left. 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. St000023The number of inner peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000183The side length of the Durfee square of an integer partition. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000352The Elizalde-Pak rank of a permutation. St000354The number of recoils of a permutation. St000386The number of factors DDU in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000646The number of big ascents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code 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. St000703The number of deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000779The tier of a permutation. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001022Number of simple modules with projective dimension 3 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. St001035The convexity degree 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. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001092The number of distinct even parts of a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over 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. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. 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. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001427The number of descents of a signed permutation. St001469The holeyness of a permutation. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001587Half of the largest even part of an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001657The number of twos in an integer partition. St001728The number of invisible descents of a permutation. St001731The factorization defect of a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. 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. St000061The number of nodes on the left branch of a binary tree. St000068The number of minimal elements in a poset. St000084The number of subtrees. St000086The number of subgraphs. St000087The number of induced subgraphs. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. 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. St000153The number of adjacent cycles of a permutation. St000159The number of distinct parts of the integer partition. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000244The cardinality of the automorphism group of a graph. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000258The burning number of a graph. St000259The diameter of a connected graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000343The number of spanning subgraphs of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000402Half the size of the symmetry class of a permutation. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000522The number of 1-protected nodes of a rooted tree. St000526The number of posets with combinatorially isomorphic order polytopes. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000537The cutwidth of a graph. St000568The hook number of a binary tree. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000700The protection number of an ordered tree. 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. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000808The number of up steps of the associated bargraph. 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. 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. St000843The decomposition number of a perfect matching. St000844The size of the largest block in the direct sum decomposition of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000903The number of different parts of an integer composition. St000912The number of maximal antichains in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St000933The number of multipartitions of sizes given by an integer partition. St000972The composition number of a graph. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001128The exponens consonantiae of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001246The maximal difference between two consecutive entries of a permutation. St001270The bandwidth of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001302The number of minimally dominating sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001432The order dimension of the partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001484The number of singletons of an integer partition. St001486The number of corners of the ribbon associated with an integer composition. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001642The Prague dimension of a graph. St001644The dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001734The lettericity of a graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001959The product of the heights of the peaks of a Dyck path. St001962The proper pathwidth of a graph. St001963The tree-depth of a graph. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000055The inversion sum of a permutation. St000081The number of edges of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000094The depth of an ordered tree. St000120The number of left 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. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000225Difference between largest and smallest parts in a partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000253The crossing number of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000292The number of ascents of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000336The leg major index of a standard tableau. 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. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000377The dinv defect of an integer partition. St000387The matching number of a graph. St000389The number of runs of ones of odd length in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000439The position of the first down step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000462The major index minus the number of excedences of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000472The sum of the ascent bottoms of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000490The intertwining number of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000523The number of 2-protected nodes of a rooted tree. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000565The major index of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. 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. 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. 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. St000624The normalized sum of the minimal distances to a greater element. St000647The number of big descents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000663The number of right floats of a permutation. St000670The reversal length of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. 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. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000761The number of ascents in an integer composition. St000778The metric dimension of a graph. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000871The number of very big ascents of a permutation. St000874The position of the last double rise in a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000891The number of distinct diagonal sums of a permutation matrix. St000897The number of different multiplicities of parts of an integer partition. St000919The number of maximal left branches of a binary tree. St000940The number of characters of the symmetric group whose value on the partition is zero. St000946The sum of the skew hook positions in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. 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. St001071The beta invariant of the graph. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001112The 3-weak dynamic number of a graph. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001153The number of blocks with even minimum in a set partition. 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. 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. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001186Number of simple modules with grade at least 3 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. St001214The aft of an integer partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of 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. St001271The competition number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001375The pancake length of a permutation. St001388The number of non-attacking neighbors of a permutation. St001391The disjunction number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. 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. 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. St001479The number of bridges of a graph. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001512The minimum rank of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001649The length of a longest trail in a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001726The number of visible inversions of 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. St001760The number of prefix or suffix reversals needed to sort a permutation. St001777The number of weak descents in an integer composition. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001801Half the number of preimage-image pairs of different parity in a permutation. 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. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001842The major index of a set partition. St001869The maximum cut size of a graph. 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). St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001931The weak major index of an integer composition regarded as a word. St001949The rigidity index of a graph. St001961The sum of the greatest common divisors of all pairs of parts. 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 $c_{n-1}$. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. 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. St000823The number of unsplittable factors of the set partition. St001568The smallest positive integer that does not appear twice in the partition. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000792The Grundy value for the game of ruler on a binary word. 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. St001114The number of odd descents of a permutation. 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. St001569The maximal modular displacement of a permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000124The cardinality of the preimage of the Simion-Schmidt map. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St000648The number of 2-excedences of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000455The second largest eigenvalue of a graph if it is integral. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001571The Cartan determinant of the integer partition. St000366The number of double descents of a permutation. St000534The number of 2-rises 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 $eA$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000913The number of ways to refine the partition into singletons. 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. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. 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. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of 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. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. 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. St000464The Schultz index of a connected graph. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000824The sum of the number of descents and the number of recoils of a permutation. St000842The breadth of a permutation. St000942The number of critical left to right maxima of the parking functions. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001487The number of inner corners of a skew partition. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St000003The number of standard Young tableaux of the partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000047The number of standard immaculate tableaux of a given shape. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000115The single entry in the last row. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000217The number of occurrences of the pattern 312 in a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000264The girth of a graph, which is not a tree. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000456The monochromatic index of a connected graph. St000517The Kreweras number of an integer partition. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000618The number of self-evacuating tableaux of given shape. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000806The semiperimeter of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. 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. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000908The length of the shortest maximal antichain in a poset. 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 $c_{n−1}$. St000993The multiplicity of the largest part of an integer partition. 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001118The acyclic chromatic index of a graph. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001182Number of indecomposable injective modules with codominant 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 $g$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001281The normalized isoperimetric number of a graph. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001470The cyclic holeyness of a permutation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001490The number of connected components of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001549The number of restricted non-inversions between exceedances. 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. St001590The crossing number of a perfect matching. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001763The Hurwitz number of an integer partition. 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. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001884The number of borders of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001933The largest multiplicity of a part in an integer partition. St001935The number of ascents in a parking function. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000017The number of inversions of a standard tableau. St000042The number of crossings of a perfect matching. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000221The number of strong fixed points of a permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 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. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000496The rcs statistic of a set partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. 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. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000787The number of flips required to make a perfect matching noncrossing. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000801The number of occurrences of the vincular pattern |312 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. St000807The sum of the heights of the valleys of the associated bargraph. St000872The number of very big descents of a permutation. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000929The constant term of the character polynomial of an integer partition. St000944The 3-degree of an integer partition. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000974The length of the trunk of an ordered tree. St000995The largest even part of an integer partition. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. 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. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001411The number of patterns 321 or 3412 in a permutation. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001513The number of nested exceedences of a permutation. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. 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. St001596The number of two-by-two squares inside a skew partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001673The degree of asymmetry of an integer composition. St001696The natural major index of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001781The interlacing number of a set partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001570The minimal number of edges to add to make a graph Hamiltonian. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000307The number of rowmotion orbits of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001964The interval resolution global dimension of a poset. St000762The sum of the positions of the weak records of an integer composition. St000914The sum of the values of the Möbius function of a poset. St000988The orbit size of a permutation under Foata's bijection. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000661The number of rises of length 3 of a Dyck path. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001520The number of strict 3-descents. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000022The number of fixed points of a permutation. St000735The last entry on the main diagonal of a standard 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. St001399The distinguishing number of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000045The number of linear extensions of a binary tree. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001890The maximum magnitude of the Möbius function of a poset. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001060The distinguishing index of a graph. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001838The number of nonempty primitive factors of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000117The number of centered tunnels of a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001910The height of the middle non-run of a Dyck path. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000916The packing number of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000247The number of singleton blocks of a set partition. St000365The number of double ascents of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001353The number of prime nodes in the modular decomposition of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001823The Stasinski-Voll length of a signed permutation. St001863The number of weak excedances of a signed permutation. St001871The number of triconnected components of a graph. St000893The number of distinct diagonal sums of an alternating sign matrix. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000048The multinomial of the parts of a partition. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000531The leading coefficient of the rook polynomial of an integer partition. St000644The number of graphs with given frequency partition. 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. 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. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001488The number of corners of a skew partition. St001595The number of standard Young tableaux of the skew partition. St001612The number of coloured multisets of cycles 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. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001875The number of simple modules with projective dimension at most 1. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001524The degree of symmetry of a binary word. St001541The Gini index 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. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. 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. 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. 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. 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. St001651The Frankl number of a lattice. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000474Dyson's crank of a partition. St000422The energy of a graph, if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000881The number of short braid edges in the graph of braid moves of a permutation. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000502The number of successions of a set partitions. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000736The last entry in the first row of a semistandard tableau. St000906The length of the shortest maximal chain in a poset. St000958The number of Bruhat factorizations of a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001889The size of the connectivity set of a signed permutation. St001937The size of the center of a parking function. St001948The number of augmented double ascents of a permutation. St000100The number of linear extensions of a poset. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000222The number of alignments in the permutation. St000236The number of cyclical small weak excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000327The number of cover relations in a poset. St000338The number of pixed points of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000633The size of the automorphism group of a poset. St000634The number of endomorphisms of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000744The length of the path to the largest entry in a standard Young tableau. St000910The number of maximal chains of minimal length in a poset. 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. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001623The number of doubly irreducible elements of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001637The number of (upper) dissectors of a poset. St001667The maximal size of a pair of weak twins for a permutation. St001668The number of points of the poset minus the width of the poset. St001705The number of occurrences of the pattern 2413 in a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001911A descent variant minus the number of inversions. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001095The number of non-isomorphic posets with precisely one further covering relation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.
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