Your data matches 364 different statistics following compositions of up to 3 maps.
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Matching statistic: St001722
(load all 31 compositions to match this statistic)
Mapping
St001722: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 = 0 + 1
1 => 1 = 0 + 1
00 => 1 = 0 + 1
01 => 1 = 0 + 1
10 => 1 = 0 + 1
11 => 1 = 0 + 1
000 => 1 = 0 + 1
001 => 1 = 0 + 1
010 => 1 = 0 + 1
011 => 1 = 0 + 1
100 => 1 = 0 + 1
101 => 1 = 0 + 1
110 => 1 = 0 + 1
111 => 1 = 0 + 1
0000 => 1 = 0 + 1
0001 => 1 = 0 + 1
0010 => 1 = 0 + 1
0011 => 1 = 0 + 1
0100 => 2 = 1 + 1
0101 => 1 = 0 + 1
0110 => 2 = 1 + 1
0111 => 1 = 0 + 1
1000 => 1 = 0 + 1
1001 => 1 = 0 + 1
1010 => 1 = 0 + 1
1011 => 1 = 0 + 1
1100 => 1 = 0 + 1
1101 => 1 = 0 + 1
1110 => 1 = 0 + 1
1111 => 1 = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000768
(load all 9 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 0
1 => [1] => 0
00 => [2] => 0
01 => [1,1] => 0
10 => [1,1] => 0
11 => [2] => 0
000 => [3] => 0
001 => [2,1] => 0
010 => [1,1,1] => 0
011 => [1,2] => 0
100 => [1,2] => 0
101 => [1,1,1] => 0
110 => [2,1] => 0
111 => [3] => 0
0000 => [4] => 0
0001 => [3,1] => 0
0010 => [2,1,1] => 0
0011 => [2,2] => 0
0100 => [1,1,2] => 0
0101 => [1,1,1,1] => 0
0110 => [1,2,1] => 1
0111 => [1,3] => 0
1000 => [1,3] => 0
1001 => [1,2,1] => 1
1010 => [1,1,1,1] => 0
1011 => [1,1,2] => 0
1100 => [2,2] => 0
1101 => [2,1,1] => 0
1110 => [3,1] => 0
1111 => [4] => 0
Description
The number of peaks in an integer composition. A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Matching statistic: St001730
(load all 17 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 0
1 => 1 => 0
00 => 01 => 0
01 => 10 => 0
10 => 11 => 0
11 => 11 => 0
000 => 001 => 0
001 => 010 => 0
010 => 101 => 0
011 => 101 => 0
100 => 101 => 0
101 => 110 => 0
110 => 111 => 0
111 => 111 => 0
0000 => 0001 => 0
0001 => 0010 => 0
0010 => 0101 => 0
0011 => 0101 => 0
0100 => 1001 => 1
0101 => 1010 => 0
0110 => 1011 => 0
0111 => 1011 => 0
1000 => 1001 => 1
1001 => 1010 => 0
1010 => 1101 => 0
1011 => 1101 => 0
1100 => 1101 => 0
1101 => 1110 => 0
1110 => 1111 => 0
1111 => 1111 => 0
Description
The number of times the path corresponding to a binary word crosses the base line. Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St000091
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 0
1 => [1] => [1] => 0
00 => [2] => [1] => 0
01 => [1,1] => [2] => 0
10 => [1,1] => [2] => 0
11 => [2] => [1] => 0
000 => [3] => [1] => 0
001 => [2,1] => [1,1] => 0
010 => [1,1,1] => [3] => 0
011 => [1,2] => [1,1] => 0
100 => [1,2] => [1,1] => 0
101 => [1,1,1] => [3] => 0
110 => [2,1] => [1,1] => 0
111 => [3] => [1] => 0
0000 => [4] => [1] => 0
0001 => [3,1] => [1,1] => 0
0010 => [2,1,1] => [1,2] => 1
0011 => [2,2] => [2] => 0
0100 => [1,1,2] => [2,1] => 0
0101 => [1,1,1,1] => [4] => 0
0110 => [1,2,1] => [1,1,1] => 0
0111 => [1,3] => [1,1] => 0
1000 => [1,3] => [1,1] => 0
1001 => [1,2,1] => [1,1,1] => 0
1010 => [1,1,1,1] => [4] => 0
1011 => [1,1,2] => [2,1] => 0
1100 => [2,2] => [2] => 0
1101 => [2,1,1] => [1,2] => 1
1110 => [3,1] => [1,1] => 0
1111 => [4] => [1] => 0
Description
The descent variation of a composition. Defined in [1].
Matching statistic: St000291
(load all 4 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0
1 => 1 => 1 => 0
00 => 01 => 01 => 0
01 => 10 => 01 => 0
10 => 11 => 11 => 0
11 => 11 => 11 => 0
000 => 001 => 001 => 0
001 => 010 => 001 => 0
010 => 101 => 011 => 0
011 => 101 => 011 => 0
100 => 101 => 011 => 0
101 => 110 => 011 => 0
110 => 111 => 111 => 0
111 => 111 => 111 => 0
0000 => 0001 => 0001 => 0
0001 => 0010 => 0001 => 0
0010 => 0101 => 0101 => 1
0011 => 0101 => 0101 => 1
0100 => 1001 => 0011 => 0
0101 => 1010 => 0011 => 0
0110 => 1011 => 0111 => 0
0111 => 1011 => 0111 => 0
1000 => 1001 => 0011 => 0
1001 => 1010 => 0011 => 0
1010 => 1101 => 0111 => 0
1011 => 1101 => 0111 => 0
1100 => 1101 => 0111 => 0
1101 => 1110 => 0111 => 0
1110 => 1111 => 1111 => 0
1111 => 1111 => 1111 => 0
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 4 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0
1 => 1 => 1 => 0
00 => 01 => 10 => 0
01 => 10 => 10 => 0
10 => 11 => 11 => 0
11 => 11 => 11 => 0
000 => 001 => 100 => 0
001 => 010 => 100 => 0
010 => 101 => 110 => 0
011 => 101 => 110 => 0
100 => 101 => 110 => 0
101 => 110 => 110 => 0
110 => 111 => 111 => 0
111 => 111 => 111 => 0
0000 => 0001 => 1000 => 0
0001 => 0010 => 1000 => 0
0010 => 0101 => 1100 => 0
0011 => 0101 => 1100 => 0
0100 => 1001 => 1010 => 1
0101 => 1010 => 1100 => 0
0110 => 1011 => 1110 => 0
0111 => 1011 => 1110 => 0
1000 => 1001 => 1010 => 1
1001 => 1010 => 1100 => 0
1010 => 1101 => 1110 => 0
1011 => 1101 => 1110 => 0
1100 => 1101 => 1110 => 0
1101 => 1110 => 1110 => 0
1110 => 1111 => 1111 => 0
1111 => 1111 => 1111 => 0
Description
The number of ascents of a binary word.
Matching statistic: St000293
(load all 4 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0
1 => 1 => 1 => 0
00 => 01 => 01 => 0
01 => 10 => 01 => 0
10 => 11 => 11 => 0
11 => 11 => 11 => 0
000 => 001 => 001 => 0
001 => 010 => 001 => 0
010 => 101 => 011 => 0
011 => 101 => 011 => 0
100 => 101 => 011 => 0
101 => 110 => 011 => 0
110 => 111 => 111 => 0
111 => 111 => 111 => 0
0000 => 0001 => 0001 => 0
0001 => 0010 => 0001 => 0
0010 => 0101 => 0101 => 1
0011 => 0101 => 0101 => 1
0100 => 1001 => 0011 => 0
0101 => 1010 => 0011 => 0
0110 => 1011 => 0111 => 0
0111 => 1011 => 0111 => 0
1000 => 1001 => 0011 => 0
1001 => 1010 => 0011 => 0
1010 => 1101 => 0111 => 0
1011 => 1101 => 0111 => 0
1100 => 1101 => 0111 => 0
1101 => 1110 => 0111 => 0
1110 => 1111 => 1111 => 0
1111 => 1111 => 1111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St000347
(load all 4 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0
1 => 1 => 1 => 0
00 => 01 => 01 => 0
01 => 10 => 01 => 0
10 => 11 => 11 => 0
11 => 11 => 11 => 0
000 => 001 => 001 => 0
001 => 010 => 001 => 0
010 => 101 => 011 => 0
011 => 101 => 011 => 0
100 => 101 => 011 => 0
101 => 110 => 011 => 0
110 => 111 => 111 => 0
111 => 111 => 111 => 0
0000 => 0001 => 0001 => 0
0001 => 0010 => 0001 => 0
0010 => 0101 => 0101 => 1
0011 => 0101 => 0101 => 1
0100 => 1001 => 0011 => 0
0101 => 1010 => 0011 => 0
0110 => 1011 => 0111 => 0
0111 => 1011 => 0111 => 0
1000 => 1001 => 0011 => 0
1001 => 1010 => 0011 => 0
1010 => 1101 => 0111 => 0
1011 => 1101 => 0111 => 0
1100 => 1101 => 0111 => 0
1101 => 1110 => 0111 => 0
1110 => 1111 => 1111 => 0
1111 => 1111 => 1111 => 0
Description
The inversion sum of a binary word. A pair $a < b$ is an inversion of a binary word $w = w_1 \cdots w_n$ if $w_a = 1 > 0 = w_b$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
Matching statistic: St000348
(load all 3 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000348: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0
1 => 1 => 1 => 0
00 => 01 => 10 => 0
01 => 10 => 10 => 0
10 => 11 => 11 => 0
11 => 11 => 11 => 0
000 => 001 => 100 => 0
001 => 010 => 100 => 0
010 => 101 => 110 => 0
011 => 101 => 110 => 0
100 => 101 => 110 => 0
101 => 110 => 110 => 0
110 => 111 => 111 => 0
111 => 111 => 111 => 0
0000 => 0001 => 1000 => 0
0001 => 0010 => 1000 => 0
0010 => 0101 => 1100 => 0
0011 => 0101 => 1100 => 0
0100 => 1001 => 1010 => 1
0101 => 1010 => 1100 => 0
0110 => 1011 => 1110 => 0
0111 => 1011 => 1110 => 0
1000 => 1001 => 1010 => 1
1001 => 1010 => 1100 => 0
1010 => 1101 => 1110 => 0
1011 => 1101 => 1110 => 0
1100 => 1101 => 1110 => 0
1101 => 1110 => 1110 => 0
1110 => 1111 => 1111 => 0
1111 => 1111 => 1111 => 0
Description
The non-inversion sum of a binary word. A pair $a < b$ is an noninversion of a binary word $w = w_1 \cdots w_n$ if $w_a < w_b$. The non-inversion sum is given by $\sum(b-a)$ over all non-inversions of $w$.
Matching statistic: St000682
(load all 3 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 => 0
1 => 1 => 1 => 0
00 => 01 => 10 => 0
01 => 10 => 10 => 0
10 => 11 => 11 => 0
11 => 11 => 11 => 0
000 => 001 => 100 => 0
001 => 010 => 100 => 0
010 => 101 => 110 => 0
011 => 101 => 110 => 0
100 => 101 => 110 => 0
101 => 110 => 110 => 0
110 => 111 => 111 => 0
111 => 111 => 111 => 0
0000 => 0001 => 1000 => 0
0001 => 0010 => 1000 => 0
0010 => 0101 => 1100 => 0
0011 => 0101 => 1100 => 0
0100 => 1001 => 1010 => 1
0101 => 1010 => 1100 => 0
0110 => 1011 => 1110 => 0
0111 => 1011 => 1110 => 0
1000 => 1001 => 1010 => 1
1001 => 1010 => 1100 => 0
1010 => 1101 => 1110 => 0
1011 => 1101 => 1110 => 0
1100 => 1101 => 1110 => 0
1101 => 1110 => 1110 => 0
1110 => 1111 => 1111 => 0
1111 => 1111 => 1111 => 0
Description
The Grundy value of Welter's game on a binary word. Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
The following 354 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. 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. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001578The minimal number of edges to add or remove to make a graph a line graph. St001871The number of triconnected components of a graph. St000047The number of standard immaculate tableaux of a given shape. St000183The side length of the Durfee square of an integer partition. St000390The number of runs of ones in a binary word. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000057The Shynar inversion number of a standard tableau. St000089The absolute variation of a composition. St000095The number of triangles of a graph. 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 [.,[[[.,.],.],. St000142The number of even parts of a partition. 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. St000185The weighted size of a partition. St000252The number of nodes of degree 3 of a binary tree. 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. St000290The major index of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in 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. St000448The number of pairs of vertices of a graph with distance 2. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000552The number of cut vertices of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000664The number of right ropes of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000807The sum of the heights of the valleys of the associated bargraph. St000877The depth of the binary word interpreted as a path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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. 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. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001308The number of induced paths on three vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. 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. St001350Half of the Albertson index of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001596The number of two-by-two squares inside a skew partition. St001623The number of doubly irreducible elements of a lattice. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001736The total number of cycles in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001846The number of elements which do not have a complement in the lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000001The number of reduced words for a 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. St000048The multinomial of the parts of a partition. St000277The number of ribbon shaped standard tableaux. 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. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000767The number of runs in an integer composition. St000805The number of peaks of the associated bargraph. 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. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $K[x]/(x^n)$. 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. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001344The neighbouring number of a permutation. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). 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. St001487The number of inner corners of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001597The Frobenius rank of a skew partition. St001732The number of peaks visible from the left. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001820The size of the image of the pop stack sorting operator. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000872The number of very big descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000963The 2-shifted major index of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001520The number of strict 3-descents. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000570The Edelman-Greene number of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001220The width of a permutation. St000260The radius of a connected graph. St000068The number of minimal elements in a poset. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000098The chromatic number of a 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. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. 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. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St000449The number of pairs of vertices of a graph with distance 4. St000637The length of the longest cycle in a graph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number 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. St001325The minimal number of occurrences of the comparability-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. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. 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. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001877Number of indecomposable injective modules with projective dimension 2. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000636The hull number of a graph. St000917The open packing number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St000447The number of pairs of vertices of a graph with distance 3. St001057The Grundy value of the game of creating an independent set in a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001691The number of kings in a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000258The burning number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000918The 2-limited packing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000264The girth of a graph, which is not a tree. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000741The Colin de Verdière graph invariant. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001625The Möbius invariant of a lattice. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001060The distinguishing index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000782The indicator function of whether a given perfect matching is an L & P matching. St000455The second largest eigenvalue of a graph if it is integral. 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. 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.