Your data matches 385 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000691
(load all 5 compositions to match this statistic)
Mapping
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => 1 = 2 - 1
10 => 1 = 2 - 1
001 => 1 = 2 - 1
010 => 2 = 3 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 2 = 3 - 1
110 => 1 = 2 - 1
0001 => 1 = 2 - 1
0010 => 2 = 3 - 1
0011 => 1 = 2 - 1
0100 => 2 = 3 - 1
0101 => 3 = 4 - 1
0110 => 2 = 3 - 1
0111 => 1 = 2 - 1
1000 => 1 = 2 - 1
1001 => 2 = 3 - 1
1010 => 3 = 4 - 1
1011 => 2 = 3 - 1
1100 => 1 = 2 - 1
1101 => 2 = 3 - 1
1110 => 1 = 2 - 1
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000847
(load all 5 compositions to match this statistic)
Mapping
St000847: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => 1 = 2 - 1
10 => 1 = 2 - 1
001 => 1 = 2 - 1
010 => 2 = 3 - 1
011 => 1 = 2 - 1
100 => 1 = 2 - 1
101 => 2 = 3 - 1
110 => 1 = 2 - 1
0001 => 1 = 2 - 1
0010 => 2 = 3 - 1
0011 => 1 = 2 - 1
0100 => 2 = 3 - 1
0101 => 3 = 4 - 1
0110 => 2 = 3 - 1
0111 => 1 = 2 - 1
1000 => 1 = 2 - 1
1001 => 2 = 3 - 1
1010 => 3 = 4 - 1
1011 => 2 = 3 - 1
1100 => 1 = 2 - 1
1101 => 2 = 3 - 1
1110 => 1 = 2 - 1
Description
The number of standard Young tableaux whose descent set is the binary word. A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation. For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
Matching statistic: St000295
(load all 5 compositions to match this statistic)
Mapping
St000295: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => 0 = 2 - 2
10 => 0 = 2 - 2
001 => 0 = 2 - 2
010 => 1 = 3 - 2
011 => 0 = 2 - 2
100 => 0 = 2 - 2
101 => 1 = 3 - 2
110 => 0 = 2 - 2
0001 => 0 = 2 - 2
0010 => 1 = 3 - 2
0011 => 0 = 2 - 2
0100 => 1 = 3 - 2
0101 => 2 = 4 - 2
0110 => 1 = 3 - 2
0111 => 0 = 2 - 2
1000 => 0 = 2 - 2
1001 => 1 = 3 - 2
1010 => 2 = 4 - 2
1011 => 1 = 3 - 2
1100 => 0 = 2 - 2
1101 => 1 = 3 - 2
1110 => 0 = 2 - 2
Description
The length of the border of a binary word. The border of a word is the longest word which is both a proper prefix and a proper suffix, including a possible empty border.
Matching statistic: St001486
(load all 4 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [2,1] => 3 = 2 + 1
10 => [1,2] => 3 = 2 + 1
001 => [3,1] => 3 = 2 + 1
010 => [2,2] => 4 = 3 + 1
011 => [2,1,1] => 3 = 2 + 1
100 => [1,3] => 3 = 2 + 1
101 => [1,2,1] => 4 = 3 + 1
110 => [1,1,2] => 3 = 2 + 1
0001 => [4,1] => 3 = 2 + 1
0010 => [3,2] => 4 = 3 + 1
0011 => [3,1,1] => 3 = 2 + 1
0100 => [2,3] => 4 = 3 + 1
0101 => [2,2,1] => 5 = 4 + 1
0110 => [2,1,2] => 4 = 3 + 1
0111 => [2,1,1,1] => 3 = 2 + 1
1000 => [1,4] => 3 = 2 + 1
1001 => [1,3,1] => 4 = 3 + 1
1010 => [1,2,2] => 5 = 4 + 1
1011 => [1,2,1,1] => 4 = 3 + 1
1100 => [1,1,3] => 3 = 2 + 1
1101 => [1,1,2,1] => 4 = 3 + 1
1110 => [1,1,1,2] => 3 = 2 + 1
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [1,1] => [1,1]
 => 2
10 => [1,1] => [1,1]
 => 2
001 => [2,1] => [2,1]
 => 2
010 => [1,1,1] => [1,1,1]
 => 3
011 => [1,2] => [2,1]
 => 2
100 => [1,2] => [2,1]
 => 2
101 => [1,1,1] => [1,1,1]
 => 3
110 => [2,1] => [2,1]
 => 2
0001 => [3,1] => [3,1]
 => 2
0010 => [2,1,1] => [2,1,1]
 => 3
0011 => [2,2] => [2,2]
 => 2
0100 => [1,1,2] => [2,1,1]
 => 3
0101 => [1,1,1,1] => [1,1,1,1]
 => 4
0110 => [1,2,1] => [2,1,1]
 => 3
0111 => [1,3] => [3,1]
 => 2
1000 => [1,3] => [3,1]
 => 2
1001 => [1,2,1] => [2,1,1]
 => 3
1010 => [1,1,1,1] => [1,1,1,1]
 => 4
1011 => [1,1,2] => [2,1,1]
 => 3
1100 => [2,2] => [2,2]
 => 2
1101 => [2,1,1] => [2,1,1]
 => 3
1110 => [3,1] => [3,1]
 => 2
Description
The length of the partition.
Matching statistic: St000011
(load all 5 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [1,1] => [1,0,1,0]
 => 2
10 => [1,1] => [1,0,1,0]
 => 2
001 => [2,1] => [1,1,0,0,1,0]
 => 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
011 => [1,2] => [1,0,1,1,0,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
110 => [2,1] => [1,1,0,0,1,0]
 => 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000015
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [1,1] => [1,0,1,0]
 => 2
10 => [1,1] => [1,0,1,0]
 => 2
001 => [2,1] => [1,1,0,0,1,0]
 => 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
011 => [1,2] => [1,0,1,1,0,0]
 => 2
100 => [1,2] => [1,0,1,1,0,0]
 => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
110 => [2,1] => [1,1,0,0,1,0]
 => 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000097
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [1,1] => ([(0,1)],2)
 => 2
10 => [1,1] => ([(0,1)],2)
 => 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
011 => [1,2] => ([(1,2)],3)
 => 2
100 => [1,2] => ([(1,2)],3)
 => 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0111 => [1,3] => ([(2,3)],4)
 => 2
1000 => [1,3] => ([(2,3)],4)
 => 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [1,1] => ([(0,1)],2)
 => 2
10 => [1,1] => ([(0,1)],2)
 => 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
011 => [1,2] => ([(1,2)],3)
 => 2
100 => [1,2] => ([(1,2)],3)
 => 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0111 => [1,3] => ([(2,3)],4)
 => 2
1000 => [1,3] => ([(2,3)],4)
 => 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
Description
The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000172
(load all 8 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
01 => [1,1] => ([(0,1)],2)
 => 2
10 => [1,1] => ([(0,1)],2)
 => 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
011 => [1,2] => ([(1,2)],3)
 => 2
100 => [1,2] => ([(1,2)],3)
 => 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0111 => [1,3] => ([(2,3)],4)
 => 2
1000 => [1,3] => ([(2,3)],4)
 => 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
Description
The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
The following 375 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000288The number of ones in a binary word. St000388The number of orbits of vertices of a graph under automorphisms. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000806The semiperimeter of the associated bargraph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. 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. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001488The number of corners of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. 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. St001812The biclique partition number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. 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. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000071The number of maximal chains in a poset. St000084The number of subtrees. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000228The size of a partition. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. 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. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000843The decomposition number of a perfect matching. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000925The number of topologically connected components of a set partition. St000926The clique-coclique number of a graph. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. 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. 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 $eA$. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001286The annihilation number of a graph. St001316The domatic number of a graph. St001330The hat guessing number 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. 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. 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. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001619The number of non-isomorphic sublattices of a lattice. 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. St001666The number of non-isomorphic subposets of a lattice which are lattices. 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. St001725The harmonious chromatic number of a graph. St001746The coalition number 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. St001949The rigidity index of a graph. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity 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. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000346The number of coarsenings of a partition. St000354The number of recoils of a permutation. St000358The number of occurrences of the pattern 31-2. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000439The position of the first down step of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000482The (zero)-forcing number of a graph. St000494The number of inversions of distance at most 3 of a permutation. St000519The largest length of a factor maximising the subword complexity. 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. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000570The Edelman-Greene number of a permutation. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000732The number of double deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000809The reduced reflection length of the permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St000864The number of circled entries of the shifted recording tableau of a permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001083The number of boxed occurrences of 132 in a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001250The number of parts of a partition that are not congruent 0 modulo 3. St001270The bandwidth of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001391The disjunction number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. 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. St001537The number of cyclic crossings of a permutation. 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. St001616The number of neutral elements in a lattice. St001644The dimension of a graph. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001720The minimal length of a chain of small intervals in a lattice. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001962The proper pathwidth of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000377The dinv defect of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001176The size of a partition minus its first part. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001638The book thickness of a graph. St001706The number of closed sets in a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a 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. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000717The number of ordinal summands of a poset. St000911The number of maximal antichains of maximal size in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000258The burning number of a graph. St000287The number of connected components of a graph. St000544The cop number of a graph. St001282The number of graphs with the same chromatic polynomial. St001363The Euler characteristic of a graph according to Knill. St001463The number of distinct columns in the nullspace of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001828The Euler characteristic of a graph. St001651The Frankl number of a lattice. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001261The Castelnuovo-Mumford regularity of a graph. St001734The lettericity of a graph. St000379The number of Hamiltonian cycles in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001871The number of triconnected components of a graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001884The number of borders of a binary word. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001479The number of bridges of a graph. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001623The number of doubly irreducible elements of a lattice. St001846The number of elements which do not have a complement in the lattice. St000264The girth of a graph, which is not a tree. St001271The competition number of a graph. St001820The size of the image of the pop stack sorting operator. St000455The second largest eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice. St001834The number of non-isomorphic minors of a graph. St001875The number of simple modules with projective dimension at most 1. St001618The cardinality of the Frattini sublattice of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000706The product of the factorials of the multiplicities of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. 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. St000929The constant term of the character polynomial of an integer partition. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.