Your data matches 385 different statistics following compositions of up to 3 maps.
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Matching statistic: St000533
(load all 12 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1
1 => [1,1] => [1,1]
 => 1
00 => [3] => [3]
 => 1
01 => [2,1] => [2,1]
 => 2
10 => [1,2] => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => 1
000 => [4] => [4]
 => 1
001 => [3,1] => [3,1]
 => 2
010 => [2,2] => [2,2]
 => 2
011 => [2,1,1] => [2,1,1]
 => 2
100 => [1,3] => [3,1]
 => 2
101 => [1,2,1] => [2,1,1]
 => 2
110 => [1,1,2] => [2,1,1]
 => 2
111 => [1,1,1,1] => [1,1,1,1]
 => 1
=> [1] => [1]
 => 1
Description
The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000760
(load all 5 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 1
1 => 1 => [1,1] => 1
00 => 00 => [3] => 1
01 => 01 => [2,1] => 2
10 => 01 => [2,1] => 2
11 => 11 => [1,1,1] => 1
000 => 000 => [4] => 1
001 => 001 => [3,1] => 2
010 => 001 => [3,1] => 2
011 => 011 => [2,1,1] => 2
100 => 001 => [3,1] => 2
101 => 011 => [2,1,1] => 2
110 => 011 => [2,1,1] => 2
111 => 111 => [1,1,1,1] => 1
=> => [1] => 1
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Matching statistic: St000767
(load all 8 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 1
1 => 1 => [1,1] => 1
00 => 00 => [3] => 1
01 => 01 => [2,1] => 2
10 => 01 => [2,1] => 2
11 => 11 => [1,1,1] => 1
000 => 000 => [4] => 1
001 => 001 => [3,1] => 2
010 => 001 => [3,1] => 2
011 => 011 => [2,1,1] => 2
100 => 001 => [3,1] => 2
101 => 011 => [2,1,1] => 2
110 => 011 => [2,1,1] => 2
111 => 111 => [1,1,1,1] => 1
=> => [1] => 1
Description
The number of runs in an integer composition. Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1]. It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
Matching statistic: St000783
(load all 12 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1
1 => [1,1] => [1,1]
 => 1
00 => [3] => [3]
 => 1
01 => [2,1] => [2,1]
 => 2
10 => [1,2] => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => 1
000 => [4] => [4]
 => 1
001 => [3,1] => [3,1]
 => 2
010 => [2,2] => [2,2]
 => 2
011 => [2,1,1] => [2,1,1]
 => 2
100 => [1,3] => [3,1]
 => 2
101 => [1,2,1] => [2,1,1]
 => 2
110 => [1,1,2] => [2,1,1]
 => 2
111 => [1,1,1,1] => [1,1,1,1]
 => 1
=> [1] => [1]
 => 1
Description
The side length of the largest staircase partition fitting into a partition. For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$. In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St000903
(load all 10 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 1
1 => 1 => [1,1] => 1
00 => 00 => [3] => 1
01 => 01 => [2,1] => 2
10 => 01 => [2,1] => 2
11 => 11 => [1,1,1] => 1
000 => 000 => [4] => 1
001 => 001 => [3,1] => 2
010 => 001 => [3,1] => 2
011 => 011 => [2,1,1] => 2
100 => 001 => [3,1] => 2
101 => 011 => [2,1,1] => 2
110 => 011 => [2,1,1] => 2
111 => 111 => [1,1,1,1] => 1
=> => [1] => 1
Description
The number of different parts of an integer composition.
Matching statistic: St000955
(load all 12 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
=> [1] => [1,0]
 => 1
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.
Matching statistic: St001432
(load all 13 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
 => 1
1 => [1,1] => [1,1]
 => 1
00 => [3] => [3]
 => 1
01 => [2,1] => [2,1]
 => 2
10 => [1,2] => [2,1]
 => 2
11 => [1,1,1] => [1,1,1]
 => 1
000 => [4] => [4]
 => 1
001 => [3,1] => [3,1]
 => 2
010 => [2,2] => [2,2]
 => 2
011 => [2,1,1] => [2,1,1]
 => 2
100 => [1,3] => [3,1]
 => 2
101 => [1,2,1] => [2,1,1]
 => 2
110 => [1,1,2] => [2,1,1]
 => 2
111 => [1,1,1,1] => [1,1,1,1]
 => 1
=> [1] => [1]
 => 1
Description
The order dimension of the partition. Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001734
(load all 13 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001734: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => 1
1 => [1,1] => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
10 => [1,2] => ([(1,2)],3)
 => 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
000 => [4] => ([],4)
 => 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
100 => [1,3] => ([(2,3)],4)
 => 2
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
=> [1] => ([],1)
 => 1
Description
The lettericity of a graph. Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$. The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.
Matching statistic: St000769
(load all 4 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 0 = 1 - 1
1 => 1 => [1,1] => 0 = 1 - 1
00 => 00 => [3] => 0 = 1 - 1
01 => 01 => [2,1] => 1 = 2 - 1
10 => 01 => [2,1] => 1 = 2 - 1
11 => 11 => [1,1,1] => 0 = 1 - 1
000 => 000 => [4] => 0 = 1 - 1
001 => 001 => [3,1] => 1 = 2 - 1
010 => 001 => [3,1] => 1 = 2 - 1
011 => 011 => [2,1,1] => 1 = 2 - 1
100 => 001 => [3,1] => 1 = 2 - 1
101 => 011 => [2,1,1] => 1 = 2 - 1
110 => 011 => [2,1,1] => 1 = 2 - 1
111 => 111 => [1,1,1,1] => 0 = 1 - 1
=> => [1] => 0 = 1 - 1
Description
The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].
Matching statistic: St001673
(load all 8 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 0 = 1 - 1
1 => 1 => [1,1] => 0 = 1 - 1
00 => 00 => [3] => 0 = 1 - 1
01 => 01 => [2,1] => 1 = 2 - 1
10 => 01 => [2,1] => 1 = 2 - 1
11 => 11 => [1,1,1] => 0 = 1 - 1
000 => 000 => [4] => 0 = 1 - 1
001 => 001 => [3,1] => 1 = 2 - 1
010 => 001 => [3,1] => 1 = 2 - 1
011 => 011 => [2,1,1] => 1 = 2 - 1
100 => 001 => [3,1] => 1 = 2 - 1
101 => 011 => [2,1,1] => 1 = 2 - 1
110 => 011 => [2,1,1] => 1 = 2 - 1
111 => 111 => [1,1,1,1] => 0 = 1 - 1
=> => [1] => 0 = 1 - 1
Description
The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.
The following 375 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000159The number of distinct parts of the integer partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000331The number of upper interactions of a Dyck path. St000388The number of orbits of vertices of a graph under automorphisms. St000390The number of runs of ones in a binary word. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000905The number of different multiplicities of parts of an integer composition. St001220The width of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001352The number of internal nodes in the modular decomposition of a graph. St001487The number of inner corners of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001569The maximal modular displacement of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. 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. St001874Lusztig's a-function for the symmetric group. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000008The major index of the composition. St000292The number of ascents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000632The jump number of the poset. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St001214The aft of an integer partition. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St000628The balance of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000983The length of the longest alternating subword. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000291The number of descents of a binary word. St000691The number of changes of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001424The number of distinct squares in a binary word. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000346The number of coarsenings of a partition. St000544The cop number of a graph. St000626The minimal period of a binary word. St000785The number of distinct colouring schemes of a graph. 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. St000822The Hadwiger number of the graph. St000982The length of the longest constant subword. St001029The size of the core of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001277The degeneracy of a graph. St001330The hat guessing number of a graph. St001358The largest degree of a regular subgraph of a graph. St001471The magnitude of a Dyck path. St001484The number of singletons of an integer partition. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000272The treewidth of a graph. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000347The inversion sum of a binary word. St000387The matching number of a graph. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001270The bandwidth 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. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001485The modular major index of a binary word. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001524The degree of symmetry of a binary word. St001592The maximal number of simple paths between any two different vertices of a graph. St001642The Prague dimension of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000258The burning number of a graph. St000259The diameter of a connected graph. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000397The Strahler number of a rooted tree. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000668The least common multiple of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps 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. St000862The number of parts of the shifted shape of a permutation. St000918The 2-limited packing number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001128The exponens consonantiae of a partition. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules 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. 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: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001488The number of corners of a skew partition. St001530The depth of a Dyck path. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001814The number of partitions interlacing the given partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000238The number of indices that are not small weak excedances. St000260The radius of a connected graph. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000348The non-inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. 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. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000549The number of odd partial sums of an integer partition. St000662The staircase size of the code of a permutation. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer 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. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. 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. St001071The beta invariant of the graph. 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. St001092The number of distinct even parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001176The size of a partition minus its first part. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001394The genus of a permutation. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001801Half the number of preimage-image pairs of different parity in a permutation. St001812The biclique partition number of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St001961The sum of the greatest common divisors of all pairs of parts. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000444The length of the maximal rise of a Dyck path. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (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. 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. 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 $eA$. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001118The acyclic chromatic index of a graph. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000806The semiperimeter of the associated bargraph. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. 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. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the 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. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000762The sum of the positions of the weak records of an integer composition. St000699The toughness times the least common multiple of 1,. St001060The distinguishing index of a graph.