There are 221 statistics on Integer partitions in the database:
(and possibly some waiting for verification)

St000003Integer partitions ⟶ ℤ
The number of standard Young tableaux of the partition.
St000010Integer partitions ⟶ ℤ
The length of the partition.
St000046Integer partitions ⟶ ℤ
The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.
St000048Integer partitions ⟶ ℤ
The multinomial of the parts of a partition.
St000049Integer partitions ⟶ ℤ
The number of set partitions whose sorted block sizes correspond to the partition.
St000063Integer partitions ⟶ ℤ
The number of linear extensions of a certain poset defined for an integer partition.
St000088Integer partitions ⟶ ℤ
The row sums of the character table of the symmetric group.
St000108Integer partitions ⟶ ℤ
The number of partitions contained in the given partition.
St000137Integer partitions ⟶ ℤ
The Grundy value of an integer partition.
St000142Integer partitions ⟶ ℤ
The number of even parts of a partition.
St000143Integer partitions ⟶ ℤ
The largest repeated part of a partition.
St000145Integer partitions ⟶ ℤ
The Dyson rank of a partition.
St000146Integer partitions ⟶ ℤ
The Andrews-Garvan crank of a partition.
St000147Integer partitions ⟶ ℤ
The largest part of an integer partition.
St000148Integer partitions ⟶ ℤ
The number of odd parts of a partition.
St000149Integer partitions ⟶ ℤ
The number of cells of the partition whose leg is zero and arm is odd.
St000150Integer partitions ⟶ ℤ
The floored half-sum of the multiplicities of a partition.
St000159Integer partitions ⟶ ℤ
The number of distinct parts of the integer partition.
St000160Integer partitions ⟶ ℤ
The multiplicity of the smallest part of a partition.
St000175Integer partitions ⟶ ℤ
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
St000179Integer partitions ⟶ ℤ
The product of the hook lengths of the integer partition.
St000182Integer partitions ⟶ ℤ
The number of permutations whose cycle type is the given integer partition.
St000183Integer partitions ⟶ ℤ
The side length of the Durfee square of an integer partition.
St000184Integer partitions ⟶ ℤ
The size of the centralizer of any permutation of given cycle type.
St000185Integer partitions ⟶ ℤ
The weighted size of a partition.
St000205Integer partitions ⟶ ℤ
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
St000206Integer partitions ⟶ ℤ
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
St000207Integer partitions ⟶ ℤ
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
St000208Integer partitions ⟶ ℤ
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
St000212Integer partitions ⟶ ℤ
The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row.
St000225Integer partitions ⟶ ℤ
Difference between largest and smallest parts in a partition.
St000228Integer partitions ⟶ ℤ
The size of a partition.
St000256Integer partitions ⟶ ℤ
The number of parts from which one can substract 2 and still get an integer partition.
St000257Integer partitions ⟶ ℤ
The number of distinct parts of a partition that occur at least twice.
St000275Integer partitions ⟶ ℤ
Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition.
St000278Integer partitions ⟶ ℤ
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
St000284Integer partitions ⟶ ℤ
The Plancherel distribution on integer partitions.
St000318Integer partitions ⟶ ℤ
The number of addable cells of the Ferrers diagram of an integer partition.
St000319Integer partitions ⟶ ℤ
The spin of an integer partition.
St000320Integer partitions ⟶ ℤ
The dinv adjustment of an integer partition.
St000321Integer partitions ⟶ ℤ
The number of integer partitions of n that are dominated by an integer partition.
St000345Integer partitions ⟶ ℤ
The number of refinements of a partition.
St000346Integer partitions ⟶ ℤ
The number of coarsenings of a partition.
St000377Integer partitions ⟶ ℤ
The dinv defect of an integer partition.
St000378Integer partitions ⟶ ℤ
The diagonal inversion number of an integer partition.
St000380Integer partitions ⟶ ℤ
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
St000384Integer partitions ⟶ ℤ
The maximal part of the shifted composition of an integer partition.
St000459Integer partitions ⟶ ℤ
The hook length of the base cell of a partition.
St000460Integer partitions ⟶ ℤ
The hook length of the last cell along the main diagonal of an integer partition.
St000473Integer partitions ⟶ ℤ
The number of parts of a partition that are strictly bigger than the number of ones.
St000474Integer partitions ⟶ ℤ
Dyson's crank of a partition.
St000475Integer partitions ⟶ ℤ
The number of parts equal to 1 in a partition.
St000477Integer partitions ⟶ ℤ
The weight of a partition according to Alladi.
St000478Integer partitions ⟶ ℤ
Another weight of a partition according to Alladi.
St000480Integer partitions ⟶ ℤ
The number of lower covers of a partition in dominance order.
St000481Integer partitions ⟶ ℤ
The number of upper covers of a partition in dominance order.
St000506Integer partitions ⟶ ℤ
The number of standard desarrangement tableaux of shape equal to the given partition.
St000509Integer partitions ⟶ ℤ
The diagonal index (content) of a partition.
St000510Integer partitions ⟶ ℤ
The number of invariant oriented cycles when acting with a permutation of given cycle type.
St000511Integer partitions ⟶ ℤ
The number of invariant subsets when acting with a permutation of given cycle type.
St000512Integer partitions ⟶ ℤ
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
St000513Integer partitions ⟶ ℤ
The number of invariant subsets of size 2 when acting with a permutation of given cycle type.
St000514Integer partitions ⟶ ℤ
The number of invariant simple graphs when acting with a permutation of given cycle type.
St000515Integer partitions ⟶ ℤ
The number of invariant set partitions when acting with a permutation of given cycle type.
St000517Integer partitions ⟶ ℤ
The Kreweras number of an integer partition.
St000531Integer partitions ⟶ ℤ
The leading coefficient of the rook polynomial of an integer partition.
St000532Integer partitions ⟶ ℤ
The total number of rook placements on a Ferrers board.
St000533Integer partitions ⟶ ℤ
The minimum of the number of parts and the size of the first part of an integer partition.
St000547Integer partitions ⟶ ℤ
The number of even non-empty partial sums of an integer partition.
St000548Integer partitions ⟶ ℤ
The number of different non-empty partial sums of an integer partition.
St000549Integer partitions ⟶ ℤ
The number of odd partial sums of an integer partition.
St000566Integer partitions ⟶ ℤ
The number of ways to select a row of a Ferrers shape and two cells in this row.
St000567Integer partitions ⟶ ℤ
The sum of the products of all pairs of parts.
St000618Integer partitions ⟶ ℤ
The number of self-evacuating tableaux of given shape.
St000620Integer partitions ⟶ ℤ
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
St000621Integer partitions ⟶ ℤ
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
St000644Integer partitions ⟶ ℤ
The number of graphs with given frequency partition.
St000667Integer partitions ⟶ ℤ
The greatest common divisor of the parts of the partition.
St000668Integer partitions ⟶ ℤ
The least common multiple of the parts of the partition.
St000681Integer partitions ⟶ ℤ
The Grundy value of Chomp on Ferrers diagrams.
St000697Integer partitions ⟶ ℤ
The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.
St000698Integer partitions ⟶ ℤ
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
St000704Integer partitions ⟶ ℤ
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
St000705Integer partitions ⟶ ℤ
The number of semistandard tableaux on a given integer partition of n with maximal entry n.
St000706Integer partitions ⟶ ℤ
The product of the factorials of the multiplicities of an integer partition.
St000707Integer partitions ⟶ ℤ
The product of the factorials of the parts.
St000708Integer partitions ⟶ ℤ
The product of the parts of an integer partition.
St000712Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 4.
St000713Integer partitions ⟶ ℤ
The dimension of the irreducible representation of Sp(4) labelled by an integer partition.
St000714Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 2.
St000715Integer partitions ⟶ ℤ
The number of semistandard Young tableaux of given shape and entries at most 3.
St000716Integer partitions ⟶ ℤ
The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
St000749Integer partitions ⟶ ℤ
The 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.
St000752Integer partitions ⟶ ℤ
The Grundy value for the game 'Couples are forever' on an integer partition.
St000755Integer partitions ⟶ ℤ
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
St000759Integer partitions ⟶ ℤ
The smallest missing part in an integer partition.
St000770Integer partitions ⟶ ℤ
The major index of an integer partition when read from bottom to top.
St000781Integer partitions ⟶ ℤ
The number of proper colouring schemes of a Ferrers diagram.
St000783Integer partitions ⟶ ℤ
The side length of the largest staircase partition fitting into a partition.
St000784Integer partitions ⟶ ℤ
The maximum of the length and the largest part of the integer partition.
St000810Integer partitions ⟶ ℤ
The 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.
St000811Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions.
St000812Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions.
St000813Integer partitions ⟶ ℤ
The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
St000814Integer partitions ⟶ ℤ
The 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.
St000815Integer partitions ⟶ ℤ
The number of semistandard Young tableaux of partition weight of given shape.
St000835Integer partitions ⟶ ℤ
The minimal difference in size when partitioning the integer partition into two subpartitions.
St000867Integer partitions ⟶ ℤ
The sum of the hook lengths in the first row of an integer partition.
St000869Integer partitions ⟶ ℤ
The sum of the hook lengths of an integer partition.
St000870Integer partitions ⟶ ℤ
The product of the hook lengths of the diagonal cells in an integer partition.
St000897Integer partitions ⟶ ℤ
The number of different multiplicities of parts of an integer partition.
St000901Integer partitions ⟶ ℤ
The cube of the number of standard Young tableaux with shape given by the partition.
St000913Integer partitions ⟶ ℤ
The number of ways to refine the partition into singletons.
St000927Integer partitions ⟶ ℤ
The alternating sum of the coefficients of the character polynomial of an integer partition.
St000928Integer partitions ⟶ ℤ
The sum of the coefficients of the character polynomial of an integer partition.
St000929Integer partitions ⟶ ℤ
The constant term of the character polynomial of an integer partition.
St000933Integer partitions ⟶ ℤ
The number of multipartitions of sizes given by an integer partition.
St000934Integer partitions ⟶ ℤ
The 2-degree of an integer partition.
St000935Integer partitions ⟶ ℤ
The number of ordered refinements of an integer partition.
St000936Integer partitions ⟶ ℤ
The number of even values of the symmetric group character corresponding to the partition.
St000937Integer partitions ⟶ ℤ
The number of positive values of the symmetric group character corresponding to the partition.
St000938Integer partitions ⟶ ℤ
The number of zeros of the symmetric group character corresponding to the partition.
St000939Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is positive.
St000940Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is zero.
St000941Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is even.
St000944Integer partitions ⟶ ℤ
The 3-degree of an integer partition.
St000992Integer partitions ⟶ ℤ
The alternating sum of the parts of an integer partition.
St000993Integer partitions ⟶ ℤ
The multiplicity of the largest part of an integer partition.
St000995Integer partitions ⟶ ℤ
The largest even part of an integer partition.
St000997Integer partitions ⟶ ℤ
The even-odd crank of an integer partition.
St001055Integer partitions ⟶ ℤ
The Grundy value for the game of removing cells of a row in an integer partition.
St001091Integer partitions ⟶ ℤ
The number of parts in an integer partition whose next smaller part has the same size.
St001092Integer partitions ⟶ ℤ
The number of distinct even parts of a partition.
St001097Integer partitions ⟶ ℤ
The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
St001098Integer partitions ⟶ ℤ
The 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 vertex labelled trees.
St001099Integer partitions ⟶ ℤ
The 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.
St001100Integer partitions ⟶ ℤ
The 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.
St001101Integer partitions ⟶ ℤ
The 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.
St001103Integer partitions ⟶ ℤ
The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123.
St001121Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition.
St001122Integer partitions ⟶ ℤ
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
St001123Integer partitions ⟶ ℤ
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
St001124Integer partitions ⟶ ℤ
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
St001127Integer partitions ⟶ ℤ
The sum of the squares of the parts of a partition.
St001128Integer partitions ⟶ ℤ
The exponens consonantiae of a partition.
St001129Integer partitions ⟶ ℤ
The product of the squares of the parts of a partition.
St001175Integer partitions ⟶ ℤ
The size of a partition minus the hook length of the base cell.
St001176Integer partitions ⟶ ℤ
The size of a partition minus its first part.
St001177Integer partitions ⟶ ℤ
Twice the mean value of the major index among all standard Young tableaux of a partition.
St001178Integer partitions ⟶ ℤ
Twelve times the variance of the major index among all standard Young tableaux of a partition.
St001214Integer partitions ⟶ ℤ
The aft of an integer partition.
St001247Integer partitions ⟶ ℤ
The number of parts of a partition that are not congruent 2 modulo 3.
St001248Integer partitions ⟶ ℤ
Sum of the even parts of a partition.
St001249Integer partitions ⟶ ℤ
Sum of the odd parts of a partition.
St001250Integer partitions ⟶ ℤ
The number of parts of a partition that are not congruent 0 modulo 3.
St001251Integer partitions ⟶ ℤ
The number of parts of a partition that are not congruent 1 modulo 3.
St001252Integer partitions ⟶ ℤ
Half the sum of the even parts of a partition.
St001262Integer partitions ⟶ ℤ
The dimension of the maximal parabolic seaweed algebra corresponding to the partition.
St001279Integer partitions ⟶ ℤ
The sum of the parts of an integer partition that are at least two.
St001280Integer partitions ⟶ ℤ
The number of parts of an integer partition that are at least two.
St001283Integer partitions ⟶ ℤ
The number of finite solvable groups that are realised by the given partition over the complex numbers.
St001284Integer partitions ⟶ ℤ
The number of finite groups that are realised by the given partition over the complex numbers.
St001360Integer partitions ⟶ ℤ
The number of covering relations in Young's lattice below a partition.
St001364Integer partitions ⟶ ℤ
The number of permutations whose cube equals a fixed permutation of given cycle type.
St001378Integer partitions ⟶ ℤ
The product of the cohook lengths of the integer partition.
St001380Integer partitions ⟶ ℤ
The number of monomer-dimer tilings of a Ferrers diagram.
St001382Integer partitions ⟶ ℤ
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
St001383Integer partitions ⟶ ℤ
The BG-rank of an integer partition.
St001384Integer partitions ⟶ ℤ
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
St001385Integer partitions ⟶ ℤ
The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition.
St001387Integer partitions ⟶ ℤ
Number of standard Young tableaux of the skew shape tracing the border of the given partition.
St001389Integer partitions ⟶ ℤ
The number of partitions of the same length below the given integer partition.
St001392Integer partitions ⟶ ℤ
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
St001400Integer partitions ⟶ ℤ
The total number of Littlewood-Richardson tableaux of given shape.
St001432Integer partitions ⟶ ℤ
The order dimension of the partition.
St001440Integer partitions ⟶ ℤ
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
St001442Integer partitions ⟶ ℤ
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
St001484Integer partitions ⟶ ℤ
The number of singletons of an integer partition.
St001525Integer partitions ⟶ ℤ
The number of symmetric hooks on the diagonal of a partition.
St001527Integer partitions ⟶ ℤ
The cyclic permutation representation number of an integer partition.
St001529Integer partitions ⟶ ℤ
The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.
St001541Integer partitions ⟶ ℤ
The Gini index of an integer partition.
St001561Integer partitions ⟶ ℤ
The value of the elementary symmetric function evaluated at 1.
St001562Integer partitions ⟶ ℤ
The value of the complete homogeneous symmetric function evaluated at 1.
St001563Integer partitions ⟶ ℤ
The value of the power-sum symmetric function evaluated at 1.
St001564Integer partitions ⟶ ℤ
The value of the forgotten symmetric functions when all variables set to 1.
St001568Integer partitions ⟶ ℤ
The smallest positive integer that does not appear twice in the partition.
St001571Integer partitions ⟶ ℤ
The Cartan determinant of the integer partition.
St001586Integer partitions ⟶ ℤ
The number of odd parts smaller than the largest even part in an integer partition.
St001587Integer partitions ⟶ ℤ
Half of the largest even part of an integer partition.
St001588Integer partitions ⟶ ℤ
The number of distinct odd parts smaller than the largest even part in an integer partition.
St001593Integer partitions ⟶ ℤ
This is the number of standard Young tableaux of the given shifted shape.
St001599Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees.
St001600Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs.
St001601Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees.
St001602Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions.
St001603Integer partitions ⟶ ℤ
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
St001604Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
St001605Integer partitions ⟶ ℤ
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
St001606Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
St001607Integer partitions ⟶ ℤ
The number of coloured graphs such that the multiplicities of colours are given by a partition.
St001608Integer partitions ⟶ ℤ
The number of coloured rooted trees such that the multiplicities of colours are given by a partition.
St001609Integer partitions ⟶ ℤ
The number of coloured trees such that the multiplicities of colours are given by a partition.
St001610Integer partitions ⟶ ℤ
The number of coloured endofunctions such that the multiplicities of colours are given by a partition.
St001611Integer partitions ⟶ ℤ
The number of multiset partitions such that the multiplicities of elements are given by a partition.
St001612Integer partitions ⟶ ℤ
The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition.
St001627Integer partitions ⟶ ℤ
The number of coloured connected graphs such that the multiplicities of colours are given by a partition.
St001628Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs.
St001657Integer partitions ⟶ ℤ
The number of twos in an integer partition.
St001659Integer partitions ⟶ ℤ
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
St001710Integer partitions ⟶ ℤ
The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation.
St001711Integer partitions ⟶ ℤ
The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.
St001714Integer partitions ⟶ ℤ
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
St001763Integer partitions ⟶ ℤ
The Hurwitz number of an integer partition.
St001767Integer partitions ⟶ ℤ
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
St001780Integer partitions ⟶ ℤ
The order of promotion on the set of standard tableaux of given shape.
St001785Integer partitions ⟶ ℤ
The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition.
St001814Integer partitions ⟶ ℤ
The number of partitions interlacing the given partition.
St001899Integer partitions ⟶ ℤ
The total number of irreducible representations contained in the higher Lie character for an integer partition.
St001900Integer partitions ⟶ ℤ
The number of distinct irreducible representations contained in the higher Lie character for an integer partition.
St001901Integer partitions ⟶ ℤ
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.