**Identifier**

**Alternating sign matrices**(21 statistics) # matrix like objects

St000065

Alternating sign matrices ⟶ ℤ

The number of entries equal to $-1$ in the alternating sign matrix.

St000066

Alternating sign matrices ⟶ ℤ

The column of the unique '1' in the first row of the alternating sign matrix.

St000067

Alternating sign matrices ⟶ ℤ

The inversion number of the alternating sign matrix.

St000076

Alternating sign matrices ⟶ ℤ

The rank of the alternating sign matrix in the alternating sign matrix poset.

St000134

Alternating sign matrices ⟶ ℤ

The size of the orbit of an alternating sign matrix under gyration.

St000187

Alternating sign matrices ⟶ ℤ

The determinant of an alternating sign matrix.

St000193

Alternating sign matrices ⟶ ℤ

The row of the unique '1' in the first column of the alternating sign matrix.

St000197

Alternating sign matrices ⟶ ℤ

The number of entries equal to positive one in the alternating sign matrix.

St000199

Alternating sign matrices ⟶ ℤ

The column of the unique '1' in the last row of the alternating sign matrix.

St000200

Alternating sign matrices ⟶ ℤ

The row of the unique '1' in the last column of the alternating sign matrix.

St000227

Alternating sign matrices ⟶ ℤ

The osculating paths major index of an alternating sign matrix.

St000332

Alternating sign matrices ⟶ ℤ

The positive inversions of an alternating sign matrix.

St000888

Alternating sign matrices ⟶ ℤ

The maximal sum of entries on a diagonal of an alternating sign matrix.

St000889

Alternating sign matrices ⟶ ℤ

The number of alternating sign matrices with the same antidiagonal sums.

St000890

Alternating sign matrices ⟶ ℤ

The number of nonzero entries in an alternating sign matrix.

St000892

Alternating sign matrices ⟶ ℤ

The maximal number of nonzero entries on a diagonal of an alternating sign matrix....

St000893

Alternating sign matrices ⟶ ℤ

The number of distinct diagonal sums of an alternating sign matrix.

St000894

Alternating sign matrices ⟶ ℤ

The trace of an alternating sign matrix.

St000895

Alternating sign matrices ⟶ ℤ

The number of ones on the main diagonal of an alternating sign matrix.

St000896

Alternating sign matrices ⟶ ℤ

The number of zeros on the main diagonal of an alternating sign matrix.

St000898

Alternating sign matrices ⟶ ℤ

The number of maximal entries in the last diagonal of the monotone triangle.

**Binary trees**(36 statistics) # tree like structures # Catalan objects # graph like objects

St000045

Binary trees ⟶ ℤ

The number of linear extensions of a binary tree.

St000050

Binary trees ⟶ ℤ

The depth/height of a binary tree.

St000051

Binary trees ⟶ ℤ

The size of the left subtree of a binary tree.

St000061

Binary trees ⟶ ℤ

The number of nodes on the left branch of a binary tree.

St000082

Binary trees ⟶ ℤ

The number of elements smaller than a binary tree in Tamari order.

St000083

Binary trees ⟶ ℤ

The number of left oriented leafs of a binary tree except the first one.

St000118

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary ....

St000121

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a bin....

St000122

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a bin....

St000125

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[[[.,.],.],.

St000126

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a....

St000127

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a....

St000128

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a....

St000129

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a....

St000130

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a....

St000131

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

St000132

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a....

St000161

Binary trees ⟶ ℤ

The sum of the sizes of the right subtrees of a binary tree.

St000196

Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [[.,.],[.,.

St000198

Binary trees ⟶ ℤ

A decimal representation of a binary tree as a code word.

St000201

Binary trees ⟶ ℤ

The number of leaf nodes in a binary tree.

St000203

Binary trees ⟶ ℤ

The number of external nodes of a binary tree.

St000204

Binary trees ⟶ ℤ

The number of internal nodes of a binary tree.

St000252

Binary trees ⟶ ℤ

The number of nodes of degree 3 of a binary tree.

St000385

Binary trees ⟶ ℤ

The number of vertices with out-degree 1 in a binary tree.

St000396

Binary trees ⟶ ℤ

The register function (or Horton-Strahler number) of a binary tree.

St000398

Binary trees ⟶ ℤ

The sum of the depths of the vertices (or total internal path length) of a binary....

St000399

Binary trees ⟶ ℤ

The external path length of a binary tree.

St000409

Binary trees ⟶ ℤ

The number of pitchforks in a binary tree.

St000411

Binary trees ⟶ ℤ

The tree factorial of a binary tree.

St000412

Binary trees ⟶ ℤ

The number of binary trees with the same underlying unordered tree.

St000414

Binary trees ⟶ ℤ

The binary logarithm of the number of binary trees with the same underlying unord....

St000568

Binary trees ⟶ ℤ

The hook number of a binary tree.

St000569

Binary trees ⟶ ℤ

The sum of the heights of the vertices of a binary tree.

St000701

Binary trees ⟶ ℤ

The protection number of a binary tree.

St000919

Binary trees ⟶ ℤ

The number of maximal left branches of a binary tree.

**Binary words**(44 statistics) # word like objects # path like objects

St000288

Binary words ⟶ ℤ

The number of ones in a binary word.

St000289

Binary words ⟶ ℤ

The decimal representation of a binary word.

St000290

Binary words ⟶ ℤ

The major index of a binary word.

St000291

Binary words ⟶ ℤ

The number of descents of a binary word.

St000292

Binary words ⟶ ℤ

The number of ascents of a binary word.

St000293

Binary words ⟶ ℤ

The number of inversions of a binary word.

St000294

Binary words ⟶ ℤ

The number of distinct factors of a binary word.

St000295

Binary words ⟶ ℤ

The length of the border of a binary word.

St000296

Binary words ⟶ ℤ

The length of the symmetric border of a binary word.

St000297

Binary words ⟶ ℤ

The number of leading ones in a binary word.

St000326

Binary words ⟶ ℤ

The position of the first one in a non-zero binary word.

St000347

Binary words ⟶ ℤ

The inversion sum of a binary word.

St000348

Binary words ⟶ ℤ

The non-inversion sum of a binary word.

St000389

Binary words ⟶ ℤ

The number of runs of ones of odd length in a binary word.

St000390

Binary words ⟶ ℤ

The number of runs of ones in a binary word.

St000391

Binary words ⟶ ℤ

The sum of the positions of the ones in a binary word.

St000392

Binary words ⟶ ℤ

The length of the longest run of ones in a binary word.

St000393

Binary words ⟶ ℤ

The number of strictly increasing runs in a binary word.

St000518

Binary words ⟶ ℤ

The number of distinct subsequences in a binary word.

St000519

Binary words ⟶ ℤ

The largest length of a factor maximising the subword complexity.

St000529

Binary words ⟶ ℤ

The number of permutations whose descent word is the given binary word.

St000543

Binary words ⟶ ℤ

The size of the conjugacy class of a binary word.

St000626

Binary words ⟶ ℤ

The minimal period of a binary word.

St000627

Binary words ⟶ ℤ

The exponent of a binary word.

St000628

Binary words ⟶ ℤ

The balance of a binary word.

St000629

Binary words ⟶ ℤ

The defect of a binary word.

St000630

Binary words ⟶ ℤ

The length of the shortest palindromic decomposition of a binary word.

St000631

Binary words ⟶ ℤ

The number of distinct palindromic decompositions of a binary word.

St000682

Binary words ⟶ ℤ

The Grundy value of Welter's game on a binary word.

St000691

Binary words ⟶ ℤ

The number of changes of a binary word.

St000753

Binary words ⟶ ℤ

The Grundy value for the game of Kayles on a binary word.

St000792

Binary words ⟶ ℤ

The Grundy value for the game of ruler on a binary word.

St000826

Binary words ⟶ ℤ

The stopping time of the decimal representation of the binary word for the 3x+1 p....

St000827

Binary words ⟶ ℤ

The decimal representation of a binary word with a leading 1.

St000847

Binary words ⟶ ℤ

The number of standard Young tableaux whose descent set is the binary word.

St000875

Binary words ⟶ ℤ

The semilength of the longest Dyck word in the Catalan factorisation of a binary ....

St000876

Binary words ⟶ ℤ

The number of factors in the Catalan decomposition of a binary word.

St000877

Binary words ⟶ ℤ

The depth of the binary word interpreted as a path.

St000878

Binary words ⟶ ℤ

The number of ones minus the number of zeros of a binary word.

St000885

Binary words ⟶ ℤ

The number of critical steps in the Catalan decomposition of a binary word.

St000921

Binary words ⟶ ℤ

The number of internal inversions of a binary word.

St000922

Binary words ⟶ ℤ

The minimal number such that all substrings of this length are unique.

St000982

Binary words ⟶ ℤ

The length of the longest constant subword.

St000983

Binary words ⟶ ℤ

The length of the longest alternating subword.

**Cores**(5 statistics) # partition like objects

**Dyck paths**(88 statistics) # Catalan objects # path like objects

St000005

Dyck paths ⟶ ℤ

The bounce statistic of a Dyck path.

St000006

Dyck paths ⟶ ℤ

The dinv statistic of a Dyck path.

St000011

Dyck paths ⟶ ℤ

The number of touch points of a Dyck path.

St000012

Dyck paths ⟶ ℤ

The area of a Dyck path.

St000013

Dyck paths ⟶ ℤ

The height of a Dyck path.

St000014

Dyck paths ⟶ ℤ

The number of parking functions supported by a Dyck path.

St000015

Dyck paths ⟶ ℤ

The number of peaks of a Dyck path.

St000024

Dyck paths ⟶ ℤ

The number of double up and double down steps of a Dyck path.

St000025

Dyck paths ⟶ ℤ

The number of initial rises of a Dyck path.

St000026

Dyck paths ⟶ ℤ

The position of the first return of a Dyck path.

St000027

Dyck paths ⟶ ℤ

The major index of a Dyck path.

St000032

Dyck paths ⟶ ℤ

The number of elements smaller than the given Dyck path in the Tamari Order.

St000038

Dyck paths ⟶ ℤ

The product of the heights of the descending steps of a Dyck path.

St000052

Dyck paths ⟶ ℤ

The number of valleys of a Dyck path not on the x-axis.

St000053

Dyck paths ⟶ ℤ

The number of valleys of the Dyck path.

St000079

Dyck paths ⟶ ℤ

The number of alternating sign matrices for a given Dyck path.

St000117

Dyck paths ⟶ ℤ

The number of centered tunnels of a Dyck path.

St000120

Dyck paths ⟶ ℤ

The number of left tunnels of a Dyck path.

St000144

Dyck paths ⟶ ℤ

The pyramid weight of the Dyck path.

St000306

Dyck paths ⟶ ℤ

The bounce count of a Dyck path.

St000329

Dyck paths ⟶ ℤ

The number of evenly positioned ascents, with the initial position equal to 1.

St000331

Dyck paths ⟶ ℤ

The number of upper interactions of a Dyck path.

St000335

Dyck paths ⟶ ℤ

The difference of lower and upper interactions.

St000340

Dyck paths ⟶ ℤ

The number of non-final maximal sub-paths of length greater than one.

St000369

Dyck paths ⟶ ℤ

The dinv deficit of a Dyck path.

St000376

Dyck paths ⟶ ℤ

The bounce deficit of a Dyck path.

St000386

Dyck paths ⟶ ℤ

The number of factors DDU in a Dyck path.

St000394

Dyck paths ⟶ ℤ

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

St000395

Dyck paths ⟶ ℤ

The sum of the heights of the peaks of a Dyck path.

St000418

Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly below a Dyck path.

St000419

Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly above a Dyck path, except for the path i....

St000420

Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly above a Dyck path.

St000421

Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly below a Dyck path, except for the path i....

St000438

Dyck paths ⟶ ℤ

The position of the last up step in a Dyck path.

St000439

Dyck paths ⟶ ℤ

The position of the first down step of a Dyck path.

St000442

Dyck paths ⟶ ℤ

The maximal area to the right of an up step of a Dyck path.

St000443

Dyck paths ⟶ ℤ

The number of long tunnels of a Dyck path.

St000444

Dyck paths ⟶ ℤ

The length of the maximal rise of a Dyck path.

St000445

Dyck paths ⟶ ℤ

The number of rises of length 1 of a Dyck path.

St000476

Dyck paths ⟶ ℤ

The sum of the semi-lengths of tunnels before a valley of a Dyck path.

St000617

Dyck paths ⟶ ℤ

The number of global maxima of a Dyck path.

St000645

Dyck paths ⟶ ℤ

The sum of the areas of the rectangles formed by two consecutive peaks and the va....

St000655

Dyck paths ⟶ ℤ

The length of the minimal rise of a Dyck path.

St000658

Dyck paths ⟶ ℤ

The number of rises of length 2 of a Dyck path.

St000659

Dyck paths ⟶ ℤ

The number of rises of length at least 2 of a Dyck path.

St000660

Dyck paths ⟶ ℤ

The number of rises of length at least 3 of a Dyck path.

St000661

Dyck paths ⟶ ℤ

The number of rises of length 3 of a Dyck path.

St000674

Dyck paths ⟶ ℤ

The number of hills of a Dyck path.

St000675

Dyck paths ⟶ ℤ

The number of centered multitunnels of a Dyck path.

St000676

Dyck paths ⟶ ℤ

The number of odd rises of a Dyck path.

St000678

Dyck paths ⟶ ℤ

The number of up steps after the last double rise of a Dyck path.

St000683

Dyck paths ⟶ ℤ

The number of points below the Dyck path such that the diagonal to the north-east....

St000684

Dyck paths ⟶ ℤ

The global dimension of the LNakayama algebra associated to a Dyck path.

St000685

Dyck paths ⟶ ℤ

The dominant dimension of the LNakayama algebra associated to a Dyck path.

St000686

Dyck paths ⟶ ℤ

The finitistic dominant dimension of a Dyck path.

St000687

Dyck paths ⟶ ℤ

The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.

St000688

Dyck paths ⟶ ℤ

The global dimension minus the dominant dimension of the LNakayama algebra associ....

St000689

Dyck paths ⟶ ℤ

The maximal n such that the minimal generator-cogenerator module in the LNakayama....

St000790

Dyck paths ⟶ ℤ

The number of pairs of centered tunnels, one strictly containing the other, of a ....

St000791

Dyck paths ⟶ ℤ

The number of pairs of left tunnels, one strictly containing the other, of a Dyck....

St000874

Dyck paths ⟶ ℤ

The position of the last double rise in a Dyck path.

St000920

Dyck paths ⟶ ℤ

The logarithmic height of a Dyck path.

St000930

Dyck paths ⟶ ℤ

The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.....

St000931

Dyck paths ⟶ ℤ

The number of occurrences of the pattern UUU in a Dyck path.

St000932

Dyck paths ⟶ ℤ

The number of occurrences of the pattern UDU in a Dyck path.

St000946

Dyck paths ⟶ ℤ

The sum of the skew hook positions in a Dyck path.

St000947

Dyck paths ⟶ ℤ

The major index of a Dyck path according to Zhao and Zhong.

St000949

Dyck paths ⟶ ℤ

Gives the number of generalised tilting modules of the corresponding LNakayama al....

St000950

Dyck paths ⟶ ℤ

Number of tilting modules of the corresponding LNakayama algebra, where a tilting....

St000951

Dyck paths ⟶ ℤ

The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra.

St000952

Dyck paths ⟶ ℤ

Gives the number of irreducible factors of the Coxeter polynomial of the Dyck pat....

St000953

Dyck paths ⟶ ℤ

The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck....

St000954

Dyck paths ⟶ ℤ

Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.

St000955

Dyck paths ⟶ ℤ

Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama a....

St000964

Dyck paths ⟶ ℤ

Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when....

St000965

Dyck paths ⟶ ℤ

The sum of the dimension of Ext^i(D(A),A) for i=1,.

St000966

Dyck paths ⟶ ℤ

Number of peaks minus the global dimension of the corresponding LNakayama algebra....

St000967

Dyck paths ⟶ ℤ

The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra....

St000968

Dyck paths ⟶ ℤ

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dy....

St000969

Dyck paths ⟶ ℤ

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dy....

St000970

Dyck paths ⟶ ℤ

Number of peaks minus the dominant dimension of the corresponding LNakayama algeb....

St000976

Dyck paths ⟶ ℤ

The sum of the positions of double up-steps of a Dyck path.

St000977

Dyck paths ⟶ ℤ

MacMahon's equal index of a Dyck path.

St000978

Dyck paths ⟶ ℤ

The sum of the positions of double down-steps of a Dyck path.

St000979

Dyck paths ⟶ ℤ

Half of MacMahon's equal index of a Dyck path.

St000980

Dyck paths ⟶ ℤ

The number of boxes weakly below the path and above the diagonal that lie below a....

St000981

Dyck paths ⟶ ℤ

The length of the longest zigzag subpath.

St000984

Dyck paths ⟶ ℤ

The number of boxes below precisely one peak.

**Finite Cartan types**(20 statistics) # Cartan type objects

St000106

Finite Cartan types ⟶ ℤ

The size of the associated Weyl group.

St000107

Finite Cartan types ⟶ ℤ

The dimension of the representation $V(\Lambda_1)$.

St000113

Finite Cartan types ⟶ ℤ

The rank of the Cartan type.

St000138

Finite Cartan types ⟶ ℤ

The Catalan number of an irreducible finite Cartan type.

St000139

Finite Cartan types ⟶ ℤ

The Coxeter number of a finite Cartan type.

St000140

Finite Cartan types ⟶ ℤ

The positive Catalan number of an irreducible finite Cartan type.

St000821

Finite Cartan types ⟶ ℤ

The determinant of the Cartan matrix.

St000851

Finite Cartan types ⟶ ℤ

The third Fuss-Catalan number of a finite Cartan type.

St000852

Finite Cartan types ⟶ ℤ

The second Fuss-Catalan number of a finite Cartan type.

St000853

Finite Cartan types ⟶ ℤ

The number of almost positive roots of a finite Cartan type.

St000854

Finite Cartan types ⟶ ℤ

The number of orbits of reflections of a finite Cartan type.

St000855

Finite Cartan types ⟶ ℤ

The number of full-support reflections in the Weyl group of a finite Cartan type.....

St000856

Finite Cartan types ⟶ ℤ

The number of conjugacy classes in the Weyl group of a finite Cartan type.

St000857

Finite Cartan types ⟶ ℤ

The number of reflections of the Weyl group of a finite Cartan type.

St000858

Finite Cartan types ⟶ ℤ

The number of factorizations of any Coxeter element into reflections of a finite ....

St000859

Finite Cartan types ⟶ ℤ

The number of parking functions of a finite Cartan type.

St000860

Finite Cartan types ⟶ ℤ

The size of the center of the Weyl group of a finite Cartan type.

St000861

Finite Cartan types ⟶ ℤ

The maximal dimension of an irreducible representation of the Weyl group of a fin....

St000865

Finite Cartan types ⟶ ℤ

The number of Coxeter elements in the Weyl group of a finite Cartan type.

St000960

Finite Cartan types ⟶ ℤ

The permanent of the Cartan matrix of a finite Cartan type.

**Gelfand-Tsetlin patterns**(11 statistics) # matrix like objects

St000072

Gelfand-Tsetlin patterns ⟶ ℤ

The number of circled entries.

St000073

Gelfand-Tsetlin patterns ⟶ ℤ

The number of boxed entries.

St000074

Gelfand-Tsetlin patterns ⟶ ℤ

The number of special entries.

St000077

Gelfand-Tsetlin patterns ⟶ ℤ

The number of boxed and circled entries.

St000114

Gelfand-Tsetlin patterns ⟶ ℤ

The sum of the entries of the Gelfand-Tsetlin pattern.

St000115

Gelfand-Tsetlin patterns ⟶ ℤ

The single entry in the last row.

St000152

Gelfand-Tsetlin patterns ⟶ ℤ

The number of boxed plus the number of special entries.

St000176

Gelfand-Tsetlin patterns ⟶ ℤ

The total number of tiles in the Gelfand-Tsetlin pattern.

St000177

Gelfand-Tsetlin patterns ⟶ ℤ

The number of free tiles in the pattern.

St000178

Gelfand-Tsetlin patterns ⟶ ℤ

Number of free entries.

St000186

Gelfand-Tsetlin patterns ⟶ ℤ

The sum of the first row in a Gelfand-Tsetlin pattern.

**Graphs**(114 statistics) # graph like objects

St000081

Graphs ⟶ ℤ

The number of edges of a graph.

St000086

Graphs ⟶ ℤ

The number of subgraphs.

St000087

Graphs ⟶ ℤ

The number of induced subgraphs.

St000093

Graphs ⟶ ℤ

The length of the maximal independent set of vertices of a graph.

St000095

Graphs ⟶ ℤ

The number of triangles of a graph.

St000096

Graphs ⟶ ℤ

The number of spanning trees of a graph.

St000097

Graphs ⟶ ℤ

The order of the largest clique of the graph.

St000098

Graphs ⟶ ℤ

The chromatic number of a graph.

St000171

Graphs ⟶ ℤ

The degree of the graph.

St000172

Graphs ⟶ ℤ

The Grundy number of a graph.

St000244

Graphs ⟶ ℤ

The cardinality of the automorphism group of a graph.

St000258

Graphs ⟶ ℤ

The burning number of a graph.

St000259

Graphs ⟶ ℤ

The diameter of a connected graph.

St000260

Graphs ⟶ ℤ

The radius of a connected graph.

St000261

Graphs ⟶ ℤ

The edge connectivity of a graph.

St000262

Graphs ⟶ ℤ

The vertex connectivity of a graph.

St000263

Graphs ⟶ ℤ

The Szeged index of a graph.

St000264

Graphs ⟶ ℤ

The girth of a graph, which is not a tree.

St000265

Graphs ⟶ ℤ

The Wiener index of a graph.

St000266

Graphs ⟶ ℤ

The number of spanning subgraphs of a graph with the same connected components.

St000267

Graphs ⟶ ℤ

The number of maximal spanning forests contained in a graph.

St000268

Graphs ⟶ ℤ

The number of strongly connected orientations of a graph.

St000269

Graphs ⟶ ℤ

The number of acyclic orientations of a graph.

St000270

Graphs ⟶ ℤ

The number of forests contained in a graph.

St000271

Graphs ⟶ ℤ

The chromatic index of a connected graph.

St000272

Graphs ⟶ ℤ

The treewidth of a graph.

St000273

Graphs ⟶ ℤ

The domination number of a graph.

St000274

Graphs ⟶ ℤ

The number of perfect matchings of a graph.

St000276

Graphs ⟶ ℤ

The size of the preimage of the map 'to graph' from Ordered trees to Graphs.

St000283

Graphs ⟶ ℤ

The size of the preimage of the map 'to graph' from Binary trees to Graphs.

St000286

Graphs ⟶ ℤ

The number of connected components of the complement of a graph.

St000287

Graphs ⟶ ℤ

The number of connected components of a graph.

St000299

Graphs ⟶ ℤ

The number of nonisomorphic vertex-induced subtrees.

St000300

Graphs ⟶ ℤ

The number of independent sets of vertices of a graph.

St000301

Graphs ⟶ ℤ

The number of facets of the stable set polytope of a graph.

St000302

Graphs ⟶ ℤ

The determinant of the distance matrix of a connected graph.

St000303

Graphs ⟶ ℤ

The determinant of the product of the incidence matrix and its transpose of a gra....

St000309

Graphs ⟶ ℤ

The number of vertices with even degree.

St000310

Graphs ⟶ ℤ

The minimal degree of a vertex of a graph.

St000311

Graphs ⟶ ℤ

The number of vertices of odd degree in a graph.

St000312

Graphs ⟶ ℤ

The number of leaves in a graph.

St000313

Graphs ⟶ ℤ

The number of degree 2 vertices of a graph.

St000315

Graphs ⟶ ℤ

The number of isolated vertices of a graph.

St000322

Graphs ⟶ ℤ

The skewness of a graph.

St000323

Graphs ⟶ ℤ

The minimal crossing number of a graph.

St000343

Graphs ⟶ ℤ

The number of spanning subgraphs of a graph.

St000344

Graphs ⟶ ℤ

The number of strongly connected outdegree sequences of a graph.

St000349

Graphs ⟶ ℤ

The number of different adjacency matrices of a graph.

St000350

Graphs ⟶ ℤ

The sum of the vertex degrees of a graph.

St000351

Graphs ⟶ ℤ

The determinant of the adjacency matrix of a graph.

St000361

Graphs ⟶ ℤ

The second Zagreb index of a graph.

St000362

Graphs ⟶ ℤ

The size of a minimal vertex cover of a graph.

St000363

Graphs ⟶ ℤ

The number of minimal vertex covers of a graph.

St000364

Graphs ⟶ ℤ

The exponent of the automorphism group of a graph.

St000368

Graphs ⟶ ℤ

The Altshuler-Steinberg determinant of a graph.

St000370

Graphs ⟶ ℤ

The genus of a graph.

St000379

Graphs ⟶ ℤ

The number of Hamiltonian cycles in a graph.

St000387

Graphs ⟶ ℤ

The matching number of a graph.

St000388

Graphs ⟶ ℤ

The number of orbits of vertices of a graph under automorphisms.

St000403

Graphs ⟶ ℤ

The Szeged index minus the Wiener index of a graph.

St000422

Graphs ⟶ ℤ

The energy of a graph, if it is integral.

St000447

Graphs ⟶ ℤ

The number of pairs of vertices of a graph with distance 3.

St000448

Graphs ⟶ ℤ

The number of pairs of vertices of a graph with distance 2.

St000449

Graphs ⟶ ℤ

The number of pairs of vertices of a graph with distance 4.

St000450

Graphs ⟶ ℤ

The number of edges minus the number of vertices plus 2 of a graph.

St000452

Graphs ⟶ ℤ

The number of distinct eigenvalues of a graph.

St000453

Graphs ⟶ ℤ

The number of distinct Laplacian eigenvalues of a graph.

St000454

Graphs ⟶ ℤ

The largest eigenvalue of a graph if it is integral.

St000455

Graphs ⟶ ℤ

The second largest eigenvalue of a graph if it is integral.

St000456

Graphs ⟶ ℤ

The monochromatic index of a connected graph.

St000464

Graphs ⟶ ℤ

The Schultz index of a connected graph.

St000465

Graphs ⟶ ℤ

The first Zagreb index of a graph.

St000466

Graphs ⟶ ℤ

The Gutman (or modified Schultz) index of a connected graph.

St000467

Graphs ⟶ ℤ

The hyper-Wiener index of a connected graph.

St000468

Graphs ⟶ ℤ

The Hosoya index of a graph.

St000469

Graphs ⟶ ℤ

The distinguishing number of a graph.

St000479

Graphs ⟶ ℤ

The Ramsey number of a graph.

St000482

Graphs ⟶ ℤ

The (zero)-forcing number of a graph.

St000535

Graphs ⟶ ℤ

The rank-width of a graph.

St000536

Graphs ⟶ ℤ

The pathwidth of a graph.

St000537

Graphs ⟶ ℤ

The cutwidth of a graph.

St000544

Graphs ⟶ ℤ

The cop number of a graph.

St000552

Graphs ⟶ ℤ

The number of cut vertices of a graph.

St000553

Graphs ⟶ ℤ

The number of blocks of a connected graph.

St000571

Graphs ⟶ ℤ

The F-index (or forgotten topological index) of a graph.

St000636

Graphs ⟶ ℤ

The hull number of a graph.

St000637

Graphs ⟶ ℤ

The length of the longest cycle in a graph.

St000671

Graphs ⟶ ℤ

The maximin edge-connectivity for choosing a subgraph.

St000699

Graphs ⟶ ℤ

The toughness times the least common multiple of 1,.

St000718

Graphs ⟶ ℤ

The largest Laplacian eigenvalue of a graph if it is integral.

St000722

Graphs ⟶ ℤ

The number of different neighbourhoods in a graph.

St000723

Graphs ⟶ ℤ

The maximal cardinality of a set of vertices with the same neighbourhood in a gra....

St000741

Graphs ⟶ ℤ

The Colin de VerdiÃ¨re graph invariant.

St000771

Graphs ⟶ ℤ

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.....

St000772

Graphs ⟶ ℤ

The multiplicity of the largest distance Laplacian eigenvalue in a connected grap....

St000773

Graphs ⟶ ℤ

The multiplicity of the largest Laplacian eigenvalue in a graph.

St000774

Graphs ⟶ ℤ

The maximal multiplicity of a Laplacian eigenvalue in a graph.

St000775

Graphs ⟶ ℤ

The multiplicity of the largest eigenvalue in a graph.

St000776

Graphs ⟶ ℤ

The maximal multiplicity of an eigenvalue in a graph.

St000777

Graphs ⟶ ℤ

The number of distinct eigenvalues of the distance Laplacian of a connected graph....

St000778

Graphs ⟶ ℤ

The metric dimension of a graph.

St000785

Graphs ⟶ ℤ

The number of distinct colouring schemes of a graph.

St000786

Graphs ⟶ ℤ

The maximal number of occurrences of a colour in a proper colouring of a graph.

St000822

Graphs ⟶ ℤ

The Hadwiger number of the graph.

St000915

Graphs ⟶ ℤ

The Ore degree of a graph.

St000916

Graphs ⟶ ℤ

The packing number of a graph.

St000917

Graphs ⟶ ℤ

The open packing number of a graph.

St000918

Graphs ⟶ ℤ

The 2-limited packing number of a graph.

St000926

Graphs ⟶ ℤ

The clique-coclique number of a graph.

St000948

Graphs ⟶ ℤ

The chromatic discriminant of a graph.

St000972

Graphs ⟶ ℤ

The composition number of a graph.

St000985

Graphs ⟶ ℤ

The number of positive eigenvalues of the adjacency matrix of the graph.

St000986

Graphs ⟶ ℤ

The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.

St000987

Graphs ⟶ ℤ

The number of positive eigenvalues of the Laplacian matrix of the graph.

**Integer compositions**(37 statistics) # partition like objects

St000008

Integer compositions ⟶ ℤ

The major index of the composition.

St000047

Integer compositions ⟶ ℤ

The number of standard immaculate tableaux of a given shape.

St000089

Integer compositions ⟶ ℤ

The absolute variation of a composition.

St000090

Integer compositions ⟶ ℤ

The variation of a composition.

St000091

Integer compositions ⟶ ℤ

The descent variation of a composition.

St000277

Integer compositions ⟶ ℤ

The size of the preimage of the map 'descent composition' from Permutations to In....

St000285

Integer compositions ⟶ ℤ

The size of the preimage of the map 'to inverse des composition' from Parking fun....

St000381

Integer compositions ⟶ ℤ

The largest part of an integer composition.

St000382

Integer compositions ⟶ ℤ

The first part of an integer composition.

St000383

Integer compositions ⟶ ℤ

The last part of an integer composition.

St000657

Integer compositions ⟶ ℤ

The smallest part of an integer composition.

St000757

Integer compositions ⟶ ℤ

The length of the longest weakly inreasing subsequence of parts of an integer com....

St000758

Integer compositions ⟶ ℤ

The length of the longest staircase fitting into an integer composition.

St000760

Integer compositions ⟶ ℤ

The length of the longest strictly decreasing subsequence of parts of an integer ....

St000761

Integer compositions ⟶ ℤ

The number of ascents in an integer composition.

St000762

Integer compositions ⟶ ℤ

The sum of the positions of the weak records of an integer composition.

St000763

Integer compositions ⟶ ℤ

The sum of the positions of the strong records of an integer composition.

St000764

Integer compositions ⟶ ℤ

The number of strong records in an integer composition.

St000765

Integer compositions ⟶ ℤ

The number of weak records in an integer composition.

St000766

Integer compositions ⟶ ℤ

The number of inversions of an integer composition.

St000767

Integer compositions ⟶ ℤ

The number of runs in an integer composition.

St000768

Integer compositions ⟶ ℤ

The number of peaks in an integer composition.

St000769

Integer compositions ⟶ ℤ

The major index of a composition.

St000805

Integer compositions ⟶ ℤ

The number of peaks of the associated bargraph.

St000806

Integer compositions ⟶ ℤ

The semiperimeter of the associated bargraph.

St000807

Integer compositions ⟶ ℤ

The sum of the heights of the valleys of the associated bargraph.

St000808

Integer compositions ⟶ ℤ

The number of up steps of the associated bargraph.

St000816

Integer compositions ⟶ ℤ

The number of standard composition tableaux of the composition.

St000817

Integer compositions ⟶ ℤ

The sum of the entries in the column specified by the composition of the change o....

St000818

Integer compositions ⟶ ℤ

The sum of the entries in the column specified by the composition of the change o....

St000820

Integer compositions ⟶ ℤ

The number of compositions obtained by rotating the composition.

St000899

Integer compositions ⟶ ℤ

The maximal number of repetitions of an integer composition.

St000900

Integer compositions ⟶ ℤ

The minimal number of repetitions of a part in an integer composition.

St000902

Integer compositions ⟶ ℤ

The minimal number of repetitions of an integer composition.

St000903

Integer compositions ⟶ ℤ

The number of different parts of an integer composition.

St000904

Integer compositions ⟶ ℤ

The maximal number of repetitions of an integer composition.

St000905

Integer compositions ⟶ ℤ

The number of different multiplicities of parts of an integer composition.

**Integer partitions**(127 statistics) # partition like objects

St000003

Integer partitions ⟶ ℤ

The number of standard Young tableaux of the partition.

St000010

Integer partitions ⟶ ℤ

The length of the partition.

St000046

Integer partitions ⟶ ℤ

The largest eigenvalue of the random to random operator acting on the simple modu....

St000048

Integer partitions ⟶ ℤ

The multinomial of the parts of a partition.

St000049

Integer partitions ⟶ ℤ

The number of set partitions whose sorted block sizes correspond to the partition....

St000063

Integer partitions ⟶ ℤ

The number of linear extensions of a certain poset defined from a partition \la....

St000088

Integer partitions ⟶ ℤ

The row sums of the character table of the symmetric group.

St000108

Integer partitions ⟶ ℤ

The number of partitions contained in the given partition.

St000137

Integer partitions ⟶ ℤ

The Grundy value of an integer partition.

St000142

Integer partitions ⟶ ℤ

The number of even parts of a partition.

St000143

Integer partitions ⟶ ℤ

The maximal repeated part of a partition.

St000145

Integer partitions ⟶ ℤ

The Dyson rank of a partition.

St000146

Integer partitions ⟶ ℤ

The Andrews-Garvan crank of a partition.

St000147

Integer partitions ⟶ ℤ

The largest part of an integer partition.

St000148

Integer partitions ⟶ ℤ

The number of odd parts of a partition.

St000149

Integer partitions ⟶ ℤ

The number of cells of the partition whose leg is zero and arm is odd.

St000150

Integer partitions ⟶ ℤ

The floored half-sum of the multiplicities of a partition.

St000159

Integer partitions ⟶ ℤ

The number of different parts of an integer partition.

St000160

Integer partitions ⟶ ℤ

Multiplicity of the smallest part of $\lambda$.

St000175

Integer partitions ⟶ ℤ

Degree of the polynomial counting the number of semistandard Young-tableaux of sh....

St000179

Integer partitions ⟶ ℤ

Product of hook lengths.

St000182

Integer partitions ⟶ ℤ

The number of permutations whose cycle type is the given integer partition.

St000183

Integer partitions ⟶ ℤ

The size of the Durfee square of an integer partition.

St000184

Integer partitions ⟶ ℤ

The size of the centralizer of any permutation of given cycle type.

St000185

Integer partitions ⟶ ℤ

The weighted size of a partition.

St000205

Integer partitions ⟶ ℤ

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and part....

St000206

Integer partitions ⟶ ℤ

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and inte....

St000207

Integer partitions ⟶ ℤ

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

St000208

Integer partitions ⟶ ℤ

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

St000212

Integer partitions ⟶ ℤ

Number of standard Young tableaux for an integer partition such that no two conse....

St000225

Integer partitions ⟶ ℤ

Difference between largest and smallest parts in a partition.

St000228

Integer partitions ⟶ ℤ

The size of a partition.

St000256

Integer partitions ⟶ ℤ

The number of parts from which one can substract 2 and still get an integer parti....

St000257

Integer partitions ⟶ ℤ

The number of distinct parts of a partition that occur at least twice.

St000275

Integer partitions ⟶ ℤ

Number of permutations whose sorted list of non zero multiplicities of the Lehmer....

St000278

Integer partitions ⟶ ℤ

The size of the preimage of the map 'to partition' from Integer compositions to I....

St000284

Integer partitions ⟶ ℤ

The Plancherel distribution on integer partitions.

St000318

Integer partitions ⟶ ℤ

The number of addable cells of the Ferrers diagram of an integer partition.

St000319

Integer partitions ⟶ ℤ

The spin of an integer partition.

St000320

Integer partitions ⟶ ℤ

The dinv adjustment of an integer partition.

St000321

Integer partitions ⟶ ℤ

The number of integer partitions of n that are dominated by an integer partition.....

St000345

Integer partitions ⟶ ℤ

The number of refinements of a partition.

St000346

Integer partitions ⟶ ℤ

The number of coarsenings of a partition.

St000377

Integer partitions ⟶ ℤ

The dinv defect of an integer partition.

St000378

Integer partitions ⟶ ℤ

The diagonal inversion number of an integer partition.

St000380

Integer partitions ⟶ ℤ

Half the perimeter of the largest rectangle that fits inside the diagram of an in....

St000384

Integer partitions ⟶ ℤ

The maximal part of the shifted composition of an integer partition.

St000459

Integer partitions ⟶ ℤ

The hook length of the base cell of a partition.

St000460

Integer partitions ⟶ ℤ

The hook length of the last cell along the main diagonal of an integer partition.....

St000473

Integer partitions ⟶ ℤ

The number of parts of a partition that are strictly bigger than the number of on....

St000474

Integer partitions ⟶ ℤ

Dyson's crank of a partition.

St000475

Integer partitions ⟶ ℤ

The number of parts equal to 1 in a partition.

St000477

Integer partitions ⟶ ℤ

The weight of a partition according to Alladi.

St000478

Integer partitions ⟶ ℤ

Another weight of a partition according to Alladi.

St000480

Integer partitions ⟶ ℤ

The number of lower covers of a partition in dominance order.

St000481

Integer partitions ⟶ ℤ

The number of upper covers of a partition in dominance order.

St000506

Integer partitions ⟶ ℤ

The number of standard desarrangement tableaux of shape equal to the given partit....

St000509

Integer partitions ⟶ ℤ

The diagonal index (content) of a partition.

St000510

Integer partitions ⟶ ℤ

The number of invariant oriented cycles when acting with a permutation of given c....

St000511

Integer partitions ⟶ ℤ

The number of invariant subsets when acting with a permutation of given cycle typ....

St000512

Integer partitions ⟶ ℤ

The number of invariant subsets of size 3 when acting with a permutation of given....

St000513

Integer partitions ⟶ ℤ

The number of invariant subsets of size 2 when acting with a permutation of given....

St000514

Integer partitions ⟶ ℤ

The number of invariant simple graphs when acting with a permutation of given cyc....

St000515

Integer partitions ⟶ ℤ

The number of invariant set partitions when acting with a permutation of given cy....

St000517

Integer partitions ⟶ ℤ

The Kreweras number of an integer partition.

St000531

Integer partitions ⟶ ℤ

The number of ways to place as many non-attacking rooks as possible on a Ferrers ....

St000532

Integer partitions ⟶ ℤ

The total number of rook placements on a Ferrers board.

St000533

Integer partitions ⟶ ℤ

The maximal number of non-attacking rooks on a Ferrers shape.

St000547

Integer partitions ⟶ ℤ

The number of even non-empty partial sums of an integer partition.

St000548

Integer partitions ⟶ ℤ

The number of different non-empty partial sums of an integer partition.

St000549

Integer partitions ⟶ ℤ

The number of odd partial sums of an integer partition.

St000566

Integer partitions ⟶ ℤ

The number of ways to select a row of a Ferrers shape and two cells in this row.

St000567

Integer partitions ⟶ ℤ

The sum of the products of all pairs of parts.

St000618

Integer partitions ⟶ ℤ

The number of self-evacuating tableaux of given shape.

St000620

Integer partitions ⟶ ℤ

The number of standard tableaux of shape equal to the given partition such that t....

St000621

Integer partitions ⟶ ℤ

The number of standard tableaux of shape equal to the given partition such that t....

St000644

Integer partitions ⟶ ℤ

The number of graphs with given frequency partition.

St000667

Integer partitions ⟶ ℤ

The greatest common divisor of the parts of the partition.

St000668

Integer partitions ⟶ ℤ

The least common multiple of the parts of the partition.

St000681

Integer partitions ⟶ ℤ

The Grundy value of Chomp on Ferrers diagrams.

St000697

Integer partitions ⟶ ℤ

The number of 3-rim hooks removed from an integer partition to obtain its associa....

St000698

Integer partitions ⟶ ℤ

The number of 2-rim hooks removed from an integer partition to obtain its associa....

St000704

Integer partitions ⟶ ℤ

The number of semistandard tableaux on a given integer partition with minimal max....

St000705

Integer partitions ⟶ ℤ

The number of semistandard tableaux on a given integer partition of n with maxima....

St000706

Integer partitions ⟶ ℤ

The product of the factorials of the multiplicities of an integer partition.

St000707

Integer partitions ⟶ ℤ

The product of the factorials of the parts.

St000708

Integer partitions ⟶ ℤ

The product of the parts of an integer partition.

St000712

Integer partitions ⟶ ℤ

The number of semistandard Young tableau of given shape, with entries at most 4.

St000713

Integer partitions ⟶ ℤ

The dimension of the irreducible representation of Sp(4) labelled by an integer p....

St000714

Integer partitions ⟶ ℤ

The number of semistandard Young tableau of given shape, with entries at most 2.

St000715

Integer partitions ⟶ ℤ

The number of semistandard Young tableau of given shape, with entries at most 3.

St000716

Integer partitions ⟶ ℤ

The dimension of the irreducible representation of Sp(6) labelled by an integer p....

St000749

Integer partitions ⟶ ℤ

The smallest integer d such that the restriction of the representation correspond....

St000752

Integer partitions ⟶ ℤ

The Grundy value for the game 'Couples are forever' on an integer partition.

St000755

Integer partitions ⟶ ℤ

The number of real roots of the characteristic polynomial of a linear recurrence ....

St000759

Integer partitions ⟶ ℤ

The smallest missing part in an integer partition.

St000770

Integer partitions ⟶ ℤ

The major index of an integer partition when read from bottom to top.

St000781

Integer partitions ⟶ ℤ

The number of proper colouring schemes of a Ferrers diagram.

St000783

Integer partitions ⟶ ℤ

The maximal number of occurrences of a colour in a proper colouring of a Ferrers ....

St000784

Integer partitions ⟶ ℤ

The maximum of the length and the number of parts of an integer partition.

St000810

Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of ....

St000811

Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of ....

St000812

Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of ....

St000813

Integer partitions ⟶ ℤ

The number of zero-one matrices with weakly decreasing column sums and row sums g....

St000814

Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of ....

St000815

Integer partitions ⟶ ℤ

The number of semistandard Young tableaux of partition weight of given shape.

St000835

Integer partitions ⟶ ℤ

The minimal difference in size when partitioning the integer partition into two s....

St000867

Integer partitions ⟶ ℤ

The sum of the hook lengths in the first column of an integer partition.

St000869

Integer partitions ⟶ ℤ

The sum of the hook lengths of an integer partition.

St000870

Integer partitions ⟶ ℤ

The product of the hook lengths of the diagonal cells in an integer partition.

St000897

Integer partitions ⟶ ℤ

The number of different multiplicities of parts of an integer partition.

St000901

Integer partitions ⟶ ℤ

The cube of the number of standard Young tableaux with shape given by the partiti....

St000913

Integer partitions ⟶ ℤ

The number of ways to refine the partition into singletons.

St000927

Integer partitions ⟶ ℤ

The alternating sum of the coefficients of the character polynomial of an integer....

St000928

Integer partitions ⟶ ℤ

The sum of the coefficients of the character polynomial of an integer partition.

St000929

Integer partitions ⟶ ℤ

The constant term of the character polynomial of an integer partition.

St000933

Integer partitions ⟶ ℤ

The number of multipartitions of sizes given by an integer partition.

St000934

Integer partitions ⟶ ℤ

The 2-degree of an integer partition.

St000935

Integer partitions ⟶ ℤ

The number of ordered refinements of an integer partition.

St000936

Integer partitions ⟶ ℤ

The number of even values of the symmetric group character corresponding to the p....

St000937

Integer partitions ⟶ ℤ

The number of positive values of the symmetric group character corresponding to t....

St000938

Integer partitions ⟶ ℤ

The number of zeros of the symmetric group character corresponding to the partiti....

St000939

Integer partitions ⟶ ℤ

The number of characters of the symmetric group whose value on the partition is p....

St000940

Integer partitions ⟶ ℤ

The number of characters of the symmetric group whose value on the partition is z....

St000941

Integer partitions ⟶ ℤ

The number of characters of the symmetric group whose value on the partition is e....

St000944

Integer partitions ⟶ ℤ

The 3-degree of an integer partition.

St000992

Integer partitions ⟶ ℤ

The alternating sum of the parts of an integer partition.

**Ordered trees**(22 statistics) # tree like structures # Catalan objects # graph like objects

St000084

Ordered trees ⟶ ℤ

The number of subtrees.

St000085

Ordered trees ⟶ ℤ

The number of linear extensions of the tree.

St000094

Ordered trees ⟶ ℤ

The depth of an ordered tree.

St000166

Ordered trees ⟶ ℤ

The depth minus 1 of an ordered tree.

St000167

Ordered trees ⟶ ℤ

The number of leaves of an ordered tree.

St000168

Ordered trees ⟶ ℤ

The number of internal nodes of an ordered tree.

St000328

Ordered trees ⟶ ℤ

The maximum number of child nodes in a tree.

St000397

Ordered trees ⟶ ℤ

The Strahler number of a rooted tree.

St000400

Ordered trees ⟶ ℤ

The path length of an ordered tree.

St000410

Ordered trees ⟶ ℤ

The tree factorial of an ordered tree.

St000413

Ordered trees ⟶ ℤ

The number of ordered trees with the same underlying unordered tree.

St000415

Ordered trees ⟶ ℤ

The size of the automorphism group of the rooted tree underlying the ordered tree....

St000416

Ordered trees ⟶ ℤ

The number of inequivalent increasing trees of an ordered tree.

St000417

Ordered trees ⟶ ℤ

The size of the automorphism group of the ordered tree.

St000521

Ordered trees ⟶ ℤ

The number of distinct subtrees of an ordered tree.

St000522

Ordered trees ⟶ ℤ

The number of 1-protected nodes of a rooted tree.

St000523

Ordered trees ⟶ ℤ

The number of 2-protected nodes of a rooted tree.

St000679

Ordered trees ⟶ ℤ

The pruning number of an ordered tree.

St000700

Ordered trees ⟶ ℤ

The protection number of an ordered tree.

St000973

Ordered trees ⟶ ℤ

The length of the boundary of an ordered tree.

St000974

Ordered trees ⟶ ℤ

The length of the trunk of an ordered tree.

St000975

Ordered trees ⟶ ℤ

The length of the boundary minus the length of the trunk of an ordered tree.

**Parking functions**(9 statistics) # word like objects # path like objects

St000135

Parking functions ⟶ ℤ

The number of lucky cars of the parking function.

St000136

Parking functions ⟶ ℤ

The dinv of a parking function.

St000165

Parking functions ⟶ ℤ

Sum of the entries.

St000188

Parking functions ⟶ ℤ

The cosum of a parking function.

St000194

Parking functions ⟶ ℤ

The number of primary dinversion pairs of a labelled dyck path corresponding to a....

St000195

Parking functions ⟶ ℤ

The number of secondary dinversion pairs of the dyck path corresponding to a park....

St000540

Parking functions ⟶ ℤ

The sum of the entries of a parking function minus its length.

St000942

Parking functions ⟶ ℤ

The number of critical left to right maxima of the parking functions.

St000943

Parking functions ⟶ ℤ

The number of spots the most unlucky car had to go further in a parking function.....

**Perfect matchings**(22 statistics) # graph like objects

St000041

Perfect matchings ⟶ ℤ

The number of nestings of a perfect matching.

St000042

Perfect matchings ⟶ ℤ

The number of crossings of a perfect matching.

St000043

Perfect matchings ⟶ ℤ

The number of crossings plus two-nestings of a perfect matching.

St000044

Perfect matchings ⟶ ℤ

The number of vertices of the unicellular map given by a perfect matching.

St000164

Perfect matchings ⟶ ℤ

The number of short pairs.

St000719

Perfect matchings ⟶ ℤ

The number of alignments in a perfect matching.

St000720

Perfect matchings ⟶ ℤ

The size of the largest partition in the oscillating tableau corresponding to the....

St000721

Perfect matchings ⟶ ℤ

The sum of the partition sizes in the oscillating tableau corresponding to a perf....

St000746

Perfect matchings ⟶ ℤ

The number of pairs with odd minimum in a perfect matching.

St000754

Perfect matchings ⟶ ℤ

The Grundy value for the game of removing nestings in a perfect matching.

St000780

Perfect matchings ⟶ ℤ

The size of the orbit under rotation of a perfect matching.

St000782

Perfect matchings ⟶ ℤ

The indicator function of whether a given perfect matching is an L & P matching.

St000787

Perfect matchings ⟶ ℤ

The number of flips required to make a perfect matching noncrossing.

St000788

Perfect matchings ⟶ ℤ

The number of nesting-similar perfect matchings of a perfect matching.

St000789

Perfect matchings ⟶ ℤ

The number of crossing-similar perfect matchings of a perfect matching.

St000819

Perfect matchings ⟶ ℤ

The propagating number of a perfect matching.

St000838

Perfect matchings ⟶ ℤ

The number of terminal right-hand endpoints when the vertices are written in orde....

St000840

Perfect matchings ⟶ ℤ

The number of closers smaller than the largest opener in a perfect matching.

St000841

Perfect matchings ⟶ ℤ

The largest opener of a perfect matching.

St000843

Perfect matchings ⟶ ℤ

The decomposition number of a perfect matching.

St000924

Perfect matchings ⟶ ℤ

The number of topologically connected components of a perfect matching.

St000945

Perfect matchings ⟶ ℤ

The number of matchings in the dihedral orbit of a perfect matching.

**Permutations**(263 statistics) # word like objects

St000001

Permutations ⟶ ℤ

The number of ways to write a permutation as a minimal length product of simple t....

St000002

Permutations ⟶ ℤ

The number of occurrences of the pattern 123 in a permutation.

St000004

Permutations ⟶ ℤ

The major index of a permutation.

St000007

Permutations ⟶ ℤ

The number of saliances of the permutation.

St000018

Permutations ⟶ ℤ

The number of inversions of a permutation.

St000019

Permutations ⟶ ℤ

The cardinality of the complement of the connectivity set.

St000020

Permutations ⟶ ℤ

The rank of the permutation.

St000021

Permutations ⟶ ℤ

The number of descents of a permutation.

St000022

Permutations ⟶ ℤ

The number of fixed points of a permutation.

St000023

Permutations ⟶ ℤ

The number of inner peaks of a permutation.

St000028

Permutations ⟶ ℤ

The number of stack-sorts needed to sort a permutation.

St000029

Permutations ⟶ ℤ

The depth of a permutation.

St000030

Permutations ⟶ ℤ

The sum of the descent differences of a permutations.

St000031

Permutations ⟶ ℤ

The number of cycles in the cycle decomposition of a permutation.

St000033

Permutations ⟶ ℤ

The number of permutations greater than or equal to the given permutation in (str....

St000034

Permutations ⟶ ℤ

The maximum defect over any reduced expression for a permutation and any subexpre....

St000035

Permutations ⟶ ℤ

The number of left outer peaks of a permutation.

St000036

Permutations ⟶ ℤ

The evaluation of the Kazhdan-Lusztig polynomial $P(id,w)$ for each permutation $w$ i....

St000037

Permutations ⟶ ℤ

The sign of a permutation.

St000039

Permutations ⟶ ℤ

The number of crossings of a permutation.

St000040

Permutations ⟶ ℤ

The number of regions of inversion arrangement of a permutation.

St000054

Permutations ⟶ ℤ

The first entry of the permutation.

St000055

Permutations ⟶ ℤ

The inversion sum of a permutation.

St000056

Permutations ⟶ ℤ

The decomposition (or block) number of a permutation.

St000058

Permutations ⟶ ℤ

The order of a permutation.

St000060

Permutations ⟶ ℤ

The greater neighbor of the maximum.

St000062

Permutations ⟶ ℤ

The length of the longest increasing subsequence of the permutation.

St000064

Permutations ⟶ ℤ

The number of one-box pattern of a permutation.

St000078

Permutations ⟶ ℤ

The number of alternating sign matrices whose left key is the permutation.

St000092

Permutations ⟶ ℤ

The number of outer peaks of a permutation.

St000099

Permutations ⟶ ℤ

The number of valleys of a permutation, including the boundary.

St000109

Permutations ⟶ ℤ

The number of elements less than or equal to the given element in Bruhat order.

St000110

Permutations ⟶ ℤ

The number of permutations less than or equal to given permutation in left weak o....

St000111

Permutations ⟶ ℤ

The sum of the descent tops (or Genocchi descents) of a permutation.

St000119

Permutations ⟶ ℤ

The number of occurrences of the pattern 321 in a permutation.

St000123

Permutations ⟶ ℤ

The difference in Coxeter length of a permutation and its image under the Simion-....

St000124

Permutations ⟶ ℤ

The cardinality of the preimage of the Simion-Schmidt map.

St000133

Permutations ⟶ ℤ

The "bounce" of a permutation.

St000141

Permutations ⟶ ℤ

The maximum drop size of a permutation.

St000153

Permutations ⟶ ℤ

The number of adjacent cycles of a permutation.

St000154

Permutations ⟶ ℤ

The sum of the descent bottoms of a permutation.

St000155

Permutations ⟶ ℤ

The number of exceedances (also excedences) of a permutation.

St000156

Permutations ⟶ ℤ

The Denert index of a permutation.

St000162

Permutations ⟶ ℤ

The number of nontrivial cycles of a permutation $\pi$ in its cycle decomposition.

St000209

Permutations ⟶ ℤ

Maximum difference of elements in cycles.

St000210

Permutations ⟶ ℤ

Minimum over maximum difference of elements in cycles.

St000213

Permutations ⟶ ℤ

The number of weak exceedances (also weak excedences) of a permutation.

St000214

Permutations ⟶ ℤ

The number of adjacencies (or small descents) of a permutation.

St000215

Permutations ⟶ ℤ

The number of adjacencies of a permutation, 0 appended.

St000216

Permutations ⟶ ℤ

The absolute length of a permutation.

St000217

Permutations ⟶ ℤ

The number of occurrences of the pattern 312 in a permutation.

St000218

Permutations ⟶ ℤ

The number of occurrences of the pattern 213 in a permutation.

St000219

Permutations ⟶ ℤ

The number of occurrences of the pattern 231 in a permutation.

St000220

Permutations ⟶ ℤ

The number of occurrences of the pattern 132 in a permutation.

St000221

Permutations ⟶ ℤ

The number of strong fixed points of a permutation.

St000222

Permutations ⟶ ℤ

The number of alignments of a permutation

St000223

Permutations ⟶ ℤ

The number of nestings of a permutation

St000224

Permutations ⟶ ℤ

The sorting index of a permutation.

St000226

Permutations ⟶ ℤ

The convexity of a permutation.

St000234

Permutations ⟶ ℤ

The number of global ascents of a permutation.

St000235

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically.

St000236

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

St000237

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i=i+1$.

St000238

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.

St000239

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \in \{i,i+1\}$.

St000240

Permutations ⟶ ℤ

The number of indices $i$ for which $\pi_i \neq i+1$.

St000241

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i = i+1$ considered cyclically.

St000242

Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically.....

St000243

Permutations ⟶ ℤ

The number of cyclic valleys and cyclic peaks of a permutation.

St000245

Permutations ⟶ ℤ

The number of ascents of a permutation.

St000246

Permutations ⟶ ℤ

The number of non-inversions of a permutation.

St000255

Permutations ⟶ ℤ

The number of reduced Kogan faces with the permutation as type.

St000279

Permutations ⟶ ℤ

The size of the preimage of the map 'cycle-as-one-line notation' from Permutation....

St000280

Permutations ⟶ ℤ

The size of the preimage of the map 'to labelling permutation' from Parking funct....

St000304

Permutations ⟶ ℤ

The load of a permutation.

St000305

Permutations ⟶ ℤ

The inverse major index of a permutation.

St000308

Permutations ⟶ ℤ

The height of the tree associated to a permutation.

St000314

Permutations ⟶ ℤ

The number of left-to-right-maxima of a permutation.

St000316

Permutations ⟶ ℤ

The number of non-left-to-right-maxima of a permutation.

St000317

Permutations ⟶ ℤ

The cycle descent number of a permutation.

St000324

Permutations ⟶ ℤ

The shape of the tree associated to a permutation.

St000325

Permutations ⟶ ℤ

The width of the tree associated to a permutation.

St000333

Permutations ⟶ ℤ

The dez statistic, the number of descents of a permutation after replacing fixed ....

St000334

Permutations ⟶ ℤ

The maz index, the major index of a permutation after replacing fixed points by z....

St000337

Permutations ⟶ ℤ

The lec statistic, the sum of the inversion numbers of the hook factors of a perm....

St000338

Permutations ⟶ ℤ

The number of pixed points of a permutation.

St000339

Permutations ⟶ ℤ

The maf index of a permutation.

St000341

Permutations ⟶ ℤ

The non-inversion sum of a permutation.

St000342

Permutations ⟶ ℤ

The cosine of a permutation.

St000352

Permutations ⟶ ℤ

The Elizalde-Pak rank of a permutation.

St000353

Permutations ⟶ ℤ

The number of inner valleys of a permutation.

St000354

Permutations ⟶ ℤ

The number of recoils of a permutation.

St000355

Permutations ⟶ ℤ

The number of occurrences of the pattern 21-3.

St000356

Permutations ⟶ ℤ

The number of occurrences of the pattern 13-2.

St000357

Permutations ⟶ ℤ

The number of occurrences of the pattern 12-3.

St000358

Permutations ⟶ ℤ

The number of occurrences of the pattern 31-2.

St000359

Permutations ⟶ ℤ

The number of occurrences of the pattern 23-1.

St000360

Permutations ⟶ ℤ

The number of occurrences of the pattern 32-1.

St000365

Permutations ⟶ ℤ

The number of double ascents of a permutation.

St000366

Permutations ⟶ ℤ

The number of double descents of a permutation.

St000367

Permutations ⟶ ℤ

The number of simsun double descents of a permutation.

St000371

Permutations ⟶ ℤ

The number of mid points of decreasing subsequences of length 3 in a permutation.....

St000372

Permutations ⟶ ℤ

The number of positions of mid points of increasing subsequences of length 3 in a....

St000373

Permutations ⟶ ℤ

The number of weak exceedences of a permutation that are also mid-points of a dec....

St000374

Permutations ⟶ ℤ

The number of exclusive right-to-left minima of a permutation.

St000375

Permutations ⟶ ℤ

The number of non weak exceedences of a permutation that are mid-points of a decr....

St000401

Permutations ⟶ ℤ

The size of the symmetry class of a permutation.

St000402

Permutations ⟶ ℤ

Half the size of the symmetry class of a permutation.

St000404

Permutations ⟶ ℤ

The number of occurrences of the pattern 3241 or of the pattern 4231 in a permuta....

St000405

Permutations ⟶ ℤ

The number of occurrences of the pattern 1324 in a permutation.

St000406

Permutations ⟶ ℤ

The number of occurrences of the pattern 3241 in a permutation.

St000407

Permutations ⟶ ℤ

The number of occurrences of the pattern 2143 in a permutation.

St000408

Permutations ⟶ ℤ

The number of occurrences of the pattern 4231 in a permutation.

St000423

Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 132 in a permutati....

St000424

Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 231 in a permutati....

St000425

Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 213 in a permutati....

St000426

Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 312 in a permutati....

St000427

Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 231 in a permutati....

St000428

Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 213 in a permutati....

St000429

Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 321 in a permutati....

St000430

Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 312 in a permutati....

St000431

Permutations ⟶ ℤ

The number of occurrences of the pattern 213 or of the pattern 321 in a permutati....

St000432

Permutations ⟶ ℤ

The number of occurrences of the pattern 231 or of the pattern 312 in a permutati....

St000433

Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 321 in a permutati....

St000434

Permutations ⟶ ℤ

The number of occurrences of the pattern 213 or of the pattern 312 in a permutati....

St000435

Permutations ⟶ ℤ

The number of occurrences of the pattern 213 or of the pattern 231 in a permutati....

St000436

Permutations ⟶ ℤ

The number of occurrences of the pattern 231 or of the pattern 321 in a permutati....

St000437

Permutations ⟶ ℤ

The number of occurrences of the pattern 312 or of the pattern 321 in a permutati....

St000440

Permutations ⟶ ℤ

The number of occurrences of the pattern 4132 or of the pattern 4231 in a permuta....

St000441

Permutations ⟶ ℤ

The number of successions (or small ascents) of a permutation.

St000446

Permutations ⟶ ℤ

The disorder of a permutation.

St000451

Permutations ⟶ ℤ

The length of the longest pattern of the form k 1 2.

St000457

Permutations ⟶ ℤ

The number of occurrences of one of the patterns 132, 213 or 321 in a permutation....

St000458

Permutations ⟶ ℤ

The number of permutations obtained by switching adjacencies or successions.

St000461

Permutations ⟶ ℤ

The rix statistic of a permutation.

St000462

Permutations ⟶ ℤ

The major index minus the number of excedences of a permutation.

St000463

Permutations ⟶ ℤ

The number of admissible inversions of a permutation.

St000470

Permutations ⟶ ℤ

The number of runs in a permutation.

St000471

Permutations ⟶ ℤ

The sum of the ascent tops of a permutation.

St000472

Permutations ⟶ ℤ

The sum of the ascent bottoms of a permutation.

St000483

Permutations ⟶ ℤ

The number of times a permutation switches from increasing to decreasing or decre....

St000484

Permutations ⟶ ℤ

The sum of St000483 over all subsequences of length at least three.

St000485

Permutations ⟶ ℤ

The length of the longest cycle of a permutation.

St000486

Permutations ⟶ ℤ

The number of cycles of length at least 3 of a permutation.

St000487

Permutations ⟶ ℤ

The length of the shortest cycle of a permutation.

St000488

Permutations ⟶ ℤ

The number of cycle of a permutation of length at most 2.

St000489

Permutations ⟶ ℤ

The number of cycle of a permutation of length at most 3.

St000494

Permutations ⟶ ℤ

The number of inversions of distance at most 3 of a permutation.

St000495

Permutations ⟶ ℤ

The number of inversions of distance at most 2 of a permutation.

St000500

Permutations ⟶ ℤ

Eigenvalues of the random-to-random operator acting on the regular representation....

St000501

Permutations ⟶ ℤ

The size of the first part in the decomposition of a permutation.

St000516

Permutations ⟶ ℤ

The number of stretching pairs of a permutation.

St000520

Permutations ⟶ ℤ

The number of patterns in a permutation.

St000530

Permutations ⟶ ℤ

The number of permutations with the same descent word as the given permutation.

St000534

Permutations ⟶ ℤ

The number of 2-rises of a permutation.

St000538

Permutations ⟶ ℤ

The number of even inversions of a permutation.

St000539

Permutations ⟶ ℤ

The number of odd inversions of a permutation.

St000541

Permutations ⟶ ℤ

The number of indices greater than or equal to 2 of a permutation such that all s....

St000542

Permutations ⟶ ℤ

The number of left-to-right-minima of a permutation.

St000545

Permutations ⟶ ℤ

The number of parabolic double cosets with minimal element being the given permut....

St000546

Permutations ⟶ ℤ

The number of global descents of a permutation.

St000570

Permutations ⟶ ℤ

The Edelman-Greene number of a permutation.

St000616

Permutations ⟶ ℤ

The inversion index of a permutation.

St000619

Permutations ⟶ ℤ

The number of cyclic descents of a permutation.

St000622

Permutations ⟶ ℤ

The number of occurrences of the patterns 2143 or 4231 in a permutation.

St000623

Permutations ⟶ ℤ

The number of occurrences of the pattern 52341 in a permutation.

St000624

Permutations ⟶ ℤ

The normalized sum of the minimal distances to a greater element.

St000625

Permutations ⟶ ℤ

The sum of the minimal distances to a greater element.

St000638

Permutations ⟶ ℤ

The number of up-down runs of a permutation.

St000646

Permutations ⟶ ℤ

The number of big ascents of a permutation.

St000647

Permutations ⟶ ℤ

The number of big descents of a permutation.

St000648

Permutations ⟶ ℤ

The number of 2-excedences of a permutation.

St000649

Permutations ⟶ ℤ

The number of 3-excedences of a permutation.

St000650

Permutations ⟶ ℤ

The number of 3-rises of a permutation.

St000651

Permutations ⟶ ℤ

The maximal size of a rise in a permutation.

St000652

Permutations ⟶ ℤ

The maximal difference between successive positions of a permutation.

St000653

Permutations ⟶ ℤ

The last descent of a permutation.

St000654

Permutations ⟶ ℤ

The first descent of a permutation.

St000662

Permutations ⟶ ℤ

The staircase size of the code of a permutation.

St000663

Permutations ⟶ ℤ

The number of right floats of a permutation.

St000664

Permutations ⟶ ℤ

The number of right ropes of a permutation.

St000665

Permutations ⟶ ℤ

The number of rafts of a permutation.

St000666

Permutations ⟶ ℤ

The number of right tethers of a permutation.

St000669

Permutations ⟶ ℤ

The number of permutations obtained by switching ascents or descents of size 2.

St000670

Permutations ⟶ ℤ

The reversal length of a permutation.

St000672

Permutations ⟶ ℤ

The number of minimal elements in Bruhat order not less than the permutation.

St000673

Permutations ⟶ ℤ

The size of the support of a permutation.

St000677

Permutations ⟶ ℤ

The standardized bi-alternating inversion number of a permutation.

St000690

Permutations ⟶ ℤ

The size of the conjugacy class of a permutation.

St000692

Permutations ⟶ ℤ

Babson and SteingrÃmsson's statistic stat of a permutation.

St000694

Permutations ⟶ ℤ

The number of affine bounded permutations that project to a given permutation.

St000696

Permutations ⟶ ℤ

The number of cycles in the breakpoint graph of a permutation.

St000702

Permutations ⟶ ℤ

The number of weak deficiencies of a permutation.

St000703

Permutations ⟶ ℤ

The number of deficiencies of a permutation.

St000709

Permutations ⟶ ℤ

The number of occurrences of 14-2-3 or 14-3-2.

St000710

Permutations ⟶ ℤ

The number of big deficiencies of a permutation.

St000711

Permutations ⟶ ℤ

The number of big exceedences of a permutation.

St000724

Permutations ⟶ ℤ

The label of the leaf of the path following the smaller label in the increasing b....

St000725

Permutations ⟶ ℤ

The smallest label of a leaf of the increasing binary tree associated to a permut....

St000726

Permutations ⟶ ℤ

The normalized sum of the leaf labels of the increasing binary tree associated to....

St000727

Permutations ⟶ ℤ

The largest label of a leaf in the binary search tree associated with the permuta....

St000731

Permutations ⟶ ℤ

The number of double exceedences of a permutation.

St000732

Permutations ⟶ ℤ

The number of double deficiencies of a permutation.

St000740

Permutations ⟶ ℤ

The last entry of a permutation.

St000742

Permutations ⟶ ℤ

The number of big ascents of a permutation after adding the value $\pi(0) = 0$.

St000750

Permutations ⟶ ℤ

The number of occurrences of the pattern 4213 in a permutation.

St000751

Permutations ⟶ ℤ

The number of occurrences of either of the pattern 2143 or 2143 in a permutation.....

St000756

Permutations ⟶ ℤ

The sum of the positions of the left to right maxima of a permutation.

St000779

Permutations ⟶ ℤ

The tier of a permutation.

St000794

Permutations ⟶ ℤ

The mak of a permutation.

St000795

Permutations ⟶ ℤ

The mad of a permutation.

St000796

Permutations ⟶ ℤ

The stat' of a permutation.

St000797

Permutations ⟶ ℤ

The stat`` of a permutation.

St000798

Permutations ⟶ ℤ

The makl of a permutation.

St000799

Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |213 in a permutation.

St000800

Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |231 in a permutation.

St000801

Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |312 in a permutation.

St000802

Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |321 in a permutation.

St000803

Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |132 in a permutation.

St000804

Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |123 in a permutation.

St000809

Permutations ⟶ ℤ

The reduced reflection length of the permutation.

St000824

Permutations ⟶ ℤ

The sum of the number of descents and the number of recoils of a permutation.

St000825

Permutations ⟶ ℤ

The sum of the number of major and the inverse major index of a permutation.

St000828

Permutations ⟶ ℤ

The spearman's rho of a permutation and the identity permutation.

St000829

Permutations ⟶ ℤ

The Ulam distance of a permutation to the identity permutation.

St000830

Permutations ⟶ ℤ

The total displacement of a permutation.

St000831

Permutations ⟶ ℤ

The number of indices that are either descents or recoils.

St000832

Permutations ⟶ ℤ

The number of permutations obtained by reversing blocks of three consecutive numb....

St000833

Permutations ⟶ ℤ

The comajor index of a permutation.

St000834

Permutations ⟶ ℤ

The number of right outer peaks of a permutation.

St000836

Permutations ⟶ ℤ

The number of descents of distance 2 of a permutation.

St000837

Permutations ⟶ ℤ

The number of ascents of distance 2 of a permutation.

St000842

Permutations ⟶ ℤ

The breadth of a permutation.

St000844

Permutations ⟶ ℤ

The size of the largest block in the direct sum decomposition of a permutation.

St000862

Permutations ⟶ ℤ

The number of parts of the shifted shape of a permutation.

St000863

Permutations ⟶ ℤ

The length of the first row of the shifted shape of a permutation.

St000864

Permutations ⟶ ℤ

The number of circled entries of the shifted recording tableau of a permutation.

St000866

Permutations ⟶ ℤ

The number of admissible inversions of a permutation in the sense of Shareshian-W....

St000868

Permutations ⟶ ℤ

The aid statistic in the sense of Shareshian-Wachs.

St000871

Permutations ⟶ ℤ

The number of very big ascents of a permutation.

St000872

Permutations ⟶ ℤ

The number of very big descents of a permutation.

St000873

Permutations ⟶ ℤ

The aix statistic of a permutation.

St000879

Permutations ⟶ ℤ

The number of long braid edges in the graph of braid moves of a permutation.

St000880

Permutations ⟶ ℤ

The number of connected components of long braid edges in the graph of braid move....

St000881

Permutations ⟶ ℤ

The number of short braid edges in the graph of braid moves of a permutation.

St000882

Permutations ⟶ ℤ

The number of connected components of short braid edges in the graph of braid mov....

St000883

Permutations ⟶ ℤ

The number of longest increasing subsequences of a permutation.

St000884

Permutations ⟶ ℤ

The number of isolated descents of a permutation.

St000886

Permutations ⟶ ℤ

The number of permutations with the same antidiagonal sums.

St000887

Permutations ⟶ ℤ

The maximal number of nonzero entries on a diagonal of a permutation matrix.

St000891

Permutations ⟶ ℤ

The number of distinct diagonal sums of a permutation matrix.

St000923

Permutations ⟶ ℤ

The minimal number with no two order isomorphic substrings of this length in a pe....

St000956

Permutations ⟶ ℤ

The maximal displacement of a permutation.

St000957

Permutations ⟶ ℤ

The number of Bruhat lower covers of a permutation.

St000958

Permutations ⟶ ℤ

The number of Bruhat factorizations of a permutation.

St000959

Permutations ⟶ ℤ

The number of strong Bruhat factorizations of a permutation.

St000961

Permutations ⟶ ℤ

The shifted major index of a permutation.

St000962

Permutations ⟶ ℤ

The 3-shifted major index of a permutation.

St000963

Permutations ⟶ ℤ

The 2-shifted major index of a permutation.

St000988

Permutations ⟶ ℤ

The orbit size of a permutation under Foata's bijection.

St000989

Permutations ⟶ ℤ

The number of final rises of a permutation.

St000990

Permutations ⟶ ℤ

The first ascent of a permutation.

St000991

Permutations ⟶ ℤ

The number of right-to-left minima of a permutation.

**Posets**(48 statistics) # graph like objects

St000068

Posets ⟶ ℤ

The number of minimal elements in a poset.

St000069

Posets ⟶ ℤ

The number of maximal elements of a poset.

St000070

Posets ⟶ ℤ

The number of antichains in a poset.

St000071

Posets ⟶ ℤ

The number of maximal chains in a poset.

St000080

Posets ⟶ ℤ

The rank of the poset.

St000100

Posets ⟶ ℤ

The number of linear extensions of a poset.

St000104

Posets ⟶ ℤ

The number of facets in the order polytope of this poset.

St000151

Posets ⟶ ℤ

The number of facets in the chain polytope of the poset.

St000180

Posets ⟶ ℤ

The number of chains of a poset.

St000181

Posets ⟶ ℤ

The number of connected components of the Hasse diagram for the poset.

St000189

Posets ⟶ ℤ

The number of elements in the poset.

St000281

Posets ⟶ ℤ

The size of the preimage of the map 'to poset' from Binary trees to Posets.

St000282

Posets ⟶ ℤ

The size of the preimage of the map 'to poset' from Ordered trees to Posets.

St000298

Posets ⟶ ℤ

The order dimension or Dushnik-Miller dimension of a poset.

St000307

Posets ⟶ ℤ

The number of rowmotion orbits of a poset.

St000327

Posets ⟶ ℤ

The number of cover relations in a poset.

St000524

Posets ⟶ ℤ

The number of posets with the same order polynomial.

St000525

Posets ⟶ ℤ

The number of posets with the same zeta polynomial.

St000526

Posets ⟶ ℤ

The number of posets with combinatorially isomorphic order polytopes.

St000527

Posets ⟶ ℤ

The width of the poset.

St000528

Posets ⟶ ℤ

The height of a poset.

St000550

Posets ⟶ ℤ

The number of modular elements of a lattice.

St000551

Posets ⟶ ℤ

The number of left modular elements of a lattice.

St000632

Posets ⟶ ℤ

The jump number of the poset.

St000633

Posets ⟶ ℤ

The size of the automorphism group of a poset.

St000634

Posets ⟶ ℤ

The number of endomorphisms of a poset.

St000635

Posets ⟶ ℤ

The number of strictly order preserving maps of a poset into itself.

St000639

Posets ⟶ ℤ

The number of relations in a poset.

St000640

Posets ⟶ ℤ

The rank of the largest boolean interval in a poset.

St000641

Posets ⟶ ℤ

The number of non-empty boolean intervals in a poset.

St000642

Posets ⟶ ℤ

The size of the smallest orbit of antichains under Panyushev complementation.

St000643

Posets ⟶ ℤ

The size of the largest orbit of antichains under Panyushev complementation.

St000656

Posets ⟶ ℤ

The number of cuts of a poset.

St000680

Posets ⟶ ℤ

The Grundy value for Hackendot on posets.

St000717

Posets ⟶ ℤ

The number of ordinal summands of a poset.

St000845

Posets ⟶ ℤ

The maximal number of elements covered by an element in a poset.

St000846

Posets ⟶ ℤ

The maximal number of elements covering an element of a poset.

St000848

Posets ⟶ ℤ

The balance constant multiplied with the number of linear extensions of a poset.

St000849

Posets ⟶ ℤ

The number of 1/3-balanced pairs in a poset.

St000850

Posets ⟶ ℤ

The number of 1/2-balanced pairs in a poset.

St000906

Posets ⟶ ℤ

The length of the shortest maximal chain in a poset.

St000907

Posets ⟶ ℤ

The number of maximal antichains of minimal length in a poset.

St000908

Posets ⟶ ℤ

The length of the shortest maximal antichain in a poset.

St000909

Posets ⟶ ℤ

The number of maximal chains of maximal size in a poset.

St000910

Posets ⟶ ℤ

The number of maximal chains of minimal length in a poset.

St000911

Posets ⟶ ℤ

The number of maximal antichains of maximal size in a poset.

St000912

Posets ⟶ ℤ

The number of maximal antichains in a poset.

St000914

Posets ⟶ ℤ

The sum of the values of the MÃ¶bius function of a poset.

**Semistandard tableaux**(11 statistics) # tableau like objects

St000101

Semistandard tableaux ⟶ ℤ

The cocharge of a semistandard tableau.

St000102

Semistandard tableaux ⟶ ℤ

The charge of a semistandard tableau.

St000103

Semistandard tableaux ⟶ ℤ

The sum of the entries of a semistandard tableau.

St000112

Semistandard tableaux ⟶ ℤ

The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....

St000116

Semistandard tableaux ⟶ ℤ

The major index of a semistandard tableau obtained by standardizing.

St000170

Semistandard tableaux ⟶ ℤ

The trace of a semistandard tableau.

St000173

Semistandard tableaux ⟶ ℤ

The segment statistic of a semistandard tableau.

St000174

Semistandard tableaux ⟶ ℤ

The flush statistic of a semistandard tableau.

St000736

Semistandard tableaux ⟶ ℤ

The last entry in the first row of a semistandard tableau.

St000737

Semistandard tableaux ⟶ ℤ

The last entry on the main diagonal of a semistandard tableau.

St000739

Semistandard tableaux ⟶ ℤ

The first entry in the last row of a semistandard tableau.

**Set partitions**(94 statistics) # partition like objects

St000105

Set partitions ⟶ ℤ

The number of blocks in the set partition.

St000163

Set partitions ⟶ ℤ

The size of the orbit of the set partition under rotation.

St000211

Set partitions ⟶ ℤ

The rank of the set partition.

St000229

Set partitions ⟶ ℤ

Sum of the difference between the maximal and the minimal elements of the blocks ....

St000230

Set partitions ⟶ ℤ

Sum of the minimal elements of the blocks of a set partition.

St000231

Set partitions ⟶ ℤ

Sum of the maximal elements of the blocks of a set partition.

St000232

Set partitions ⟶ ℤ

The number of crossings of a set partition.

St000233

Set partitions ⟶ ℤ

The number of nestings of a set partition.

St000247

Set partitions ⟶ ℤ

The number of singleton blocks of a set partition.

St000248

Set partitions ⟶ ℤ

The number of anti-singletons of a set partition.

St000249

Set partitions ⟶ ℤ

St000250

Set partitions ⟶ ℤ

St000251

Set partitions ⟶ ℤ

The number of nonsingleton blocks of a set partition.

St000253

Set partitions ⟶ ℤ

The crossing number of a set partition.

St000254

Set partitions ⟶ ℤ

The nesting number of a set partition.

St000490

Set partitions ⟶ ℤ

The intertwining number of a set partition.

St000491

Set partitions ⟶ ℤ

The number of inversions of a set partition.

St000492

Set partitions ⟶ ℤ

The rob statistic of a set partition.

St000493

Set partitions ⟶ ℤ

The los statistic of a set partition.

St000496

Set partitions ⟶ ℤ

The rcs statistic of a set partition.

St000497

Set partitions ⟶ ℤ

The lcb statistic of a set partition.

St000498

Set partitions ⟶ ℤ

The lcs statistic of a set partition.

St000499

Set partitions ⟶ ℤ

The rcb statistic of a set partition.

St000502

Set partitions ⟶ ℤ

The number of successions of a set partitions.

St000503

Set partitions ⟶ ℤ

The maximal difference between two elements in a common block.

St000504

Set partitions ⟶ ℤ

The cardinality of the first block of a set partition.

St000505

Set partitions ⟶ ℤ

The biggest entry in the block containing the 1.

St000554

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2},{3}} in a set partition.

St000555

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} in a set partition.

St000556

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} in a set partition.

St000557

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} in a set partition.

St000558

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2}} in a set partition.

St000559

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.

St000560

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.

St000561

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2,3}} in a set partition.

St000562

Set partitions ⟶ ℤ

The number of internal points of a set partition.

St000563

Set partitions ⟶ ℤ

The number of overlapping pairs of blocks of a set partition.

St000564

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} in a set partition.

St000565

Set partitions ⟶ ℤ

The major index of a set partition.

St000572

Set partitions ⟶ ℤ

The dimension exponent of a set partition.

St000573

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 ....

St000574

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a ....

St000575

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element ....

St000576

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a ....

St000577

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.....

St000578

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.

St000579

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.....

St000580

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is m....

St000581

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is ma....

St000582

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is ma....

St000583

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1,2 ar....

St000584

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is m....

St000585

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) a....

St000586

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.

St000587

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.

St000588

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 i....

St000589

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) a....

St000590

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is ma....

St000591

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.

St000592

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.

St000593

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.

St000594

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3....

St000595

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.

St000596

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is m....

St000597

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) a....

St000598

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is....

St000599

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive....

St000600

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) a....

St000601

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3....

St000602

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.

St000603

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.

St000604

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is m....

St000605

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) a....

St000606

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3....

St000607

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is ma....

St000608

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 i....

St000609

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

St000610

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.

St000611

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.

St000612

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) a....

St000613

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is ma....

St000614

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is ma....

St000615

Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.

St000695

Set partitions ⟶ ℤ

The number of blocks in the first part of the atomic decomposition of a set parti....

St000728

Set partitions ⟶ ℤ

The dimension of a set partition.

St000729

Set partitions ⟶ ℤ

The minimal arc length of a set partition.

St000730

Set partitions ⟶ ℤ

The maximal arc length of a set partition.

St000747

Set partitions ⟶ ℤ

A variant of the major index of a set partition.

St000748

Set partitions ⟶ ℤ

The major index of the permutation obtained by flattening the set partition.

St000793

Set partitions ⟶ ℤ

The length of the longest partition in the vacillating tableau corresponding to a....

St000823

Set partitions ⟶ ℤ

The number of unsplittable factors of the set partition.

St000839

Set partitions ⟶ ℤ

The largest opener of a set partition.

St000925

Set partitions ⟶ ℤ

The number of topologically connected components of a set partition.

St000971

Set partitions ⟶ ℤ

The smallest closer of a set partition.

**Standard tableaux**(20 statistics) # tableau like objects

St000009

Standard tableaux ⟶ ℤ

The charge of a standard tableau.

St000016

Standard tableaux ⟶ ℤ

The number of attacking pairs of a standard tableau.

St000017

Standard tableaux ⟶ ℤ

The number of inversions of a standard tableau.

St000057

Standard tableaux ⟶ ℤ

The Shynar inversion number of a standard tableau.

St000059

Standard tableaux ⟶ ℤ

The inversion number of a standard tableau as defined by Haglund and Stevens.

St000075

Standard tableaux ⟶ ℤ

The orbit size of a standard tableau under promotion.

St000157

Standard tableaux ⟶ ℤ

The number of descents of a standard tableau.

St000169

Standard tableaux ⟶ ℤ

The cocharge of a standard tableau.

St000330

Standard tableaux ⟶ ℤ

The (standard) major index of a standard tableau.

St000336

Standard tableaux ⟶ ℤ

The leg major index of a standard tableau.

St000507

Standard tableaux ⟶ ℤ

The number of ascents of a standard tableau.

St000508

Standard tableaux ⟶ ℤ

Eigenvalues of the random-to-random operator acting on a simple module.

St000693

Standard tableaux ⟶ ℤ

The modular (standard) major index of a standard tableau.

St000733

Standard tableaux ⟶ ℤ

The row containing the largest entry of a standard tableau.

St000734

Standard tableaux ⟶ ℤ

The last entry in the first row of a standard tableau.

St000735

Standard tableaux ⟶ ℤ

The last entry on the main diagonal of a standard tableau.

St000738

Standard tableaux ⟶ ℤ

The first entry in the last row of a standard tableau.

St000743

Standard tableaux ⟶ ℤ

The number of entries in a standard Young tableau such that the next integer is a....

St000744

Standard tableaux ⟶ ℤ

The length of the path to the largest entry in a standard Young tableau.

St000745

Standard tableaux ⟶ ℤ

The index of the last row whose first entry is the row number in a standard Young....