Identifier
There are 992 statistics in the database, and possibly some more waiting for verification:
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
St000158
Cores ⟶ ℤ
The length of a core.
St000190
Cores ⟶ ℤ
The size of a core.
St000191
Cores ⟶ ℤ
The number of strong covers of a core.
St000192
Cores ⟶ ℤ
Number of covers of a core in weak Bruhat order.
St000202
Cores ⟶ ℤ
The number of $k$-cores contained by a $k$-core.
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 Verdière 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 Steingrímsson'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 Möbius 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
The number of singletons (St000247) plus the number of antisingletons (St000248) ....
St000250
Set partitions ⟶ ℤ
The number of blocks (St000105) plus the number of antisingletons (St000248) of a....
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....