Your data matches 414 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000679
Mapping
St000679: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 1
[[],[]]
 => 1
[[[]]]
 => 1
[[],[],[]]
 => 1
[[],[[]]]
 => 1
[[[]],[]]
 => 1
[[[],[]]]
 => 2
[[[[]]]]
 => 1
[[],[],[],[]]
 => 1
[[],[],[[]]]
 => 1
[[],[[]],[]]
 => 1
[[],[[],[]]]
 => 2
[[],[[[]]]]
 => 1
[[[]],[],[]]
 => 1
[[[]],[[]]]
 => 1
[[[],[]],[]]
 => 2
[[[[]]],[]]
 => 1
[[[],[],[]]]
 => 2
[[[],[[]]]]
 => 2
[[[[]],[]]]
 => 2
[[[[],[]]]]
 => 2
[[[[[]]]]]
 => 1
Description
The pruning number of an ordered tree. A hanging branch of an ordered tree is a proper factor of the form $[^r]^r$ for some $r\geq 1$. A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch. A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to $[]$.
Matching statistic: St000201
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 1
[[],[]]
 => [[.,.],.]
 => 1
[[[]]]
 => [.,[.,.]]
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000396
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 1
[[],[]]
 => [[.,.],.]
 => 1
[[[]]]
 => [.,[.,.]]
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 2
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 2
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 2
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 2
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 2
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 1
Description
The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000920
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000920: 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]
 => 1
[[[]]]
 => [1,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 2
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St001503
(load all 9 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001503: 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]
 => 1
[[[]]]
 => [1,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 2
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 2
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 1
Description
The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
Matching statistic: St000196
(load all 3 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => 0 = 1 - 1
[[],[]]
 => [[.,.],.]
 => 0 = 1 - 1
[[[]]]
 => [.,[.,.]]
 => 0 = 1 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => 0 = 1 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => 1 = 2 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => 0 = 1 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => 0 = 1 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => 0 = 1 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => 0 = 1 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => 1 = 2 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => 1 = 2 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => 0 = 1 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => 1 = 2 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => 0 = 1 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => 0 = 1 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => 0 = 1 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => 1 = 2 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => 0 = 1 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => 0 = 1 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000386
(load all 11 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000386: 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]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 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
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001022
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001022: 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]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 0 = 1 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001037
(load all 9 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001037: 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]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000068
(load all 2 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => ([],1)
 => 1
[[],[]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1
[[[]]]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1
[[[],[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2
[[[[]]]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
Description
The number of minimal elements in a poset.
The following 404 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000071The number of maximal chains in a poset. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000659The number of rises of length at least 2 of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000862The number of parts of the shifted shape of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001597The Frobenius rank of a skew partition. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St000023The number of inner peaks of a permutation. 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. 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. St000397The Strahler number of a rooted tree. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000650The number of 3-rises of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001394The genus of a permutation. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St000010The length of the partition. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000325The width of the tree associated to a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000346The number of coarsenings of a partition. St000352The Elizalde-Pak rank of a permutation. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St000409The number of pitchforks in a binary tree. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000805The number of peaks 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. St000822The Hadwiger number of the graph. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000919The number of maximal left branches of a binary tree. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000996The number of exclusive left-to-right maxima of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001116The game chromatic number of a graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001261The Castelnuovo-Mumford regularity of a graph. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001280The number of parts of an integer partition that are at least two. 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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001358The largest degree of a regular subgraph of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001792The arboricity of a graph. St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001928The number of non-overlapping descents in a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001963The tree-depth of a graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000062The length of the longest increasing subsequence of the permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000147The largest part of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000223The number of nestings in the permutation. St000252The number of nodes of degree 3 of a binary tree. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000308The height of the tree associated to a permutation. St000317The cycle descent number of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000353The number of inner valleys of a permutation. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000387The matching number of a graph. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000516The number of stretching pairs of a permutation. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. 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. St001115The number of even descents of a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. 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. St001393The induced matching number of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001471The magnitude of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001565The number of arithmetic progressions of length 2 in a permutation. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001644The dimension of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 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. St001712The number of natural descents of a standard Young tableau. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001797The number of overfull subgraphs of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001812The biclique partition number of a graph. St001871The number of triconnected components of a graph. St001874Lusztig's a-function for the symmetric group. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000640The rank of the largest boolean interval in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000486The number of cycles of length at least 3 of a permutation. St000711The number of big exceedences of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000031The number of cycles in the cycle decomposition of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St001820The size of the image of the pop stack sorting operator. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000291The number of descents of a binary word. St000480The number of lower covers of a partition in dominance order. St000539The number of odd inversions 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. St000661The number of rises of length 3 of a Dyck path. St000710The number of big deficiencies of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000938The number of zeros of the symmetric group character corresponding to the partition. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001592The maximal number of simple paths between any two different vertices of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001487The number of inner corners of a skew partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000531The leading coefficient of the rook polynomial of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. 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. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000454The largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St001964The interval resolution global dimension of a poset. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001626The number of maximal proper sublattices of a lattice. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St000422The energy of a graph, if it is integral. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001877Number of indecomposable injective modules with projective dimension 2. 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. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St000741The Colin de Verdière graph invariant. St001625The Möbius invariant of a lattice. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001413Half the length of the longest even length palindromic prefix of a binary word. St001481The minimal height of a peak of a Dyck path. St001949The rigidity index of a graph. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000906The length of the shortest maximal chain in a poset. St001130The number of two successive successions in a permutation. 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. 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. St001352The number of internal nodes in the modular decomposition of a graph. St001705The number of occurrences of the pattern 2413 in 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001118The acyclic chromatic index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001490The number of connected components of a skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000879The number of long braid edges in the graph of braid moves of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001645The pebbling number of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001828The Euler characteristic of a graph. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001948The number of augmented double ascents of a permutation.