Your data matches 228 different statistics following compositions of up to 3 maps.
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Matching statistic: St000922
(load all 47 compositions to match this statistic)
Mapping
St000922: 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 => 2 = 1 + 1
01 => 1 = 0 + 1
10 => 1 = 0 + 1
11 => 2 = 1 + 1
000 => 3 = 2 + 1
001 => 2 = 1 + 1
010 => 2 = 1 + 1
011 => 2 = 1 + 1
100 => 2 = 1 + 1
101 => 2 = 1 + 1
110 => 2 = 1 + 1
111 => 3 = 2 + 1
Description
The minimal number such that all substrings of this length are unique.
Matching statistic: St000393
(load all 19 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 0 + 1
1 => 1 => 1 = 0 + 1
00 => 00 => 2 = 1 + 1
01 => 01 => 1 = 0 + 1
10 => 01 => 1 = 0 + 1
11 => 11 => 2 = 1 + 1
000 => 000 => 3 = 2 + 1
001 => 001 => 2 = 1 + 1
010 => 001 => 2 = 1 + 1
011 => 011 => 2 = 1 + 1
100 => 001 => 2 = 1 + 1
101 => 011 => 2 = 1 + 1
110 => 011 => 2 = 1 + 1
111 => 111 => 3 = 2 + 1
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000982
(load all 13 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 0 + 1
1 => 1 => 1 = 0 + 1
00 => 00 => 2 = 1 + 1
01 => 01 => 1 = 0 + 1
10 => 01 => 1 = 0 + 1
11 => 11 => 2 = 1 + 1
000 => 000 => 3 = 2 + 1
001 => 001 => 2 = 1 + 1
010 => 001 => 2 = 1 + 1
011 => 011 => 2 = 1 + 1
100 => 001 => 2 = 1 + 1
101 => 011 => 2 = 1 + 1
110 => 011 => 2 = 1 + 1
111 => 111 => 3 = 2 + 1
Description
The length of the longest constant subword.
Matching statistic: St001416
(load all 23 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001416: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 0 + 1
1 => 1 => 1 = 0 + 1
00 => 00 => 2 = 1 + 1
01 => 01 => 1 = 0 + 1
10 => 01 => 1 = 0 + 1
11 => 11 => 2 = 1 + 1
000 => 000 => 3 = 2 + 1
001 => 001 => 2 = 1 + 1
010 => 001 => 2 = 1 + 1
011 => 011 => 2 = 1 + 1
100 => 001 => 2 = 1 + 1
101 => 011 => 2 = 1 + 1
110 => 011 => 2 = 1 + 1
111 => 111 => 3 = 2 + 1
Description
The length of a longest palindromic factor of a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.
Matching statistic: St001417
(load all 23 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 = 0 + 1
1 => 1 => 1 = 0 + 1
00 => 00 => 2 = 1 + 1
01 => 01 => 1 = 0 + 1
10 => 01 => 1 = 0 + 1
11 => 11 => 2 = 1 + 1
000 => 000 => 3 = 2 + 1
001 => 001 => 2 = 1 + 1
010 => 001 => 2 = 1 + 1
011 => 011 => 2 = 1 + 1
100 => 001 => 2 = 1 + 1
101 => 011 => 2 = 1 + 1
110 => 011 => 2 = 1 + 1
111 => 111 => 3 = 2 + 1
Description
The length of a longest palindromic subword of a binary word. A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword. Any binary word of length $n$ contains a palindromic subword of length at least $n/2$, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with $n/2$ zeros and ending with $n/2$ ones.
Matching statistic: St000295
(load all 2 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000295: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 00 => 1
01 => 01 => 10 => 0
10 => 01 => 10 => 0
11 => 11 => 11 => 1
000 => 000 => 000 => 2
001 => 001 => 010 => 1
010 => 001 => 010 => 1
011 => 011 => 101 => 1
100 => 001 => 010 => 1
101 => 011 => 101 => 1
110 => 011 => 101 => 1
111 => 111 => 111 => 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: St000377
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => [2]
 => 0
1 => ([(0,1)],2)
 => [2]
 => 0
00 => ([(0,2),(2,1)],3)
 => [3]
 => 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 0
11 => ([(0,2),(2,1)],3)
 => [3]
 => 1
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 1
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 2
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000691
(load all 9 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 01 => 1
01 => 10 => 11 => 0
10 => 10 => 11 => 0
11 => 11 => 10 => 1
000 => 000 => 010 => 2
001 => 100 => 110 => 1
010 => 100 => 110 => 1
011 => 110 => 100 => 1
100 => 100 => 110 => 1
101 => 110 => 100 => 1
110 => 110 => 100 => 1
111 => 111 => 101 => 2
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: St000145
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
 => [2]
 => 1 = 0 + 1
1 => ([(0,1)],2)
 => [2]
 => 1 = 0 + 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 1 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
11 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 1 + 1
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 2 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2 = 1 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2 = 1 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2 = 1 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2 = 1 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2 = 1 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2 = 1 + 1
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 2 + 1
Description
The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000381
(load all 2 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => 1 = 0 + 1
1 => 1 => [1] => 1 = 0 + 1
00 => 00 => [2] => 2 = 1 + 1
01 => 01 => [1,1] => 1 = 0 + 1
10 => 01 => [1,1] => 1 = 0 + 1
11 => 11 => [2] => 2 = 1 + 1
000 => 000 => [3] => 3 = 2 + 1
001 => 001 => [2,1] => 2 = 1 + 1
010 => 001 => [2,1] => 2 = 1 + 1
011 => 011 => [1,2] => 2 = 1 + 1
100 => 001 => [2,1] => 2 = 1 + 1
101 => 011 => [1,2] => 2 = 1 + 1
110 => 011 => [1,2] => 2 = 1 + 1
111 => 111 => [3] => 3 = 2 + 1
Description
The largest part of an integer composition.
The following 218 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000626The minimal period of a binary word. St000676The number of odd rises of a Dyck path. St000808The number of up steps of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000983The length of the longest alternating subword. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001884The number of borders of a binary word. St000384The maximal part of the shifted composition of an integer partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. 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. St000340The number of non-final maximal constant sub-paths of length greater than one. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000651The maximal size of a rise in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000941The number of characters of the symmetric group whose value on the partition is even. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001638The book thickness of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000443The number of long tunnels of a Dyck path. St000482The (zero)-forcing number of a graph. St000553The number of blocks of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000746The number of pairs with odd minimum in a perfect matching. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000935The number of ordered refinements of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001270The bandwidth of a graph. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001530The depth of a Dyck path. St001644The dimension of a graph. St001668The number of points of the poset minus the width of the poset. St001733The number of weak left to right maxima of a Dyck path. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000144The pyramid weight of the Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000784The maximum of the length and the largest part of the integer partition. St000863The length of the first row of the shifted shape of a permutation. St000912The number of maximal antichains in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. 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. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St000444The length of the maximal rise of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000045The number of linear extensions of a binary tree. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000284The Plancherel distribution on integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000460The hook length of the last cell along the main diagonal of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the 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. St001592The maximal number of simple paths between any two different vertices of a graph. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer 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. St000219The number of occurrences of the pattern 231 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral.