**Identifier**

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

St000065
The number of entries equal to $-1$ in the alternating sign matrix.

St000066 The column of the unique '1' in the first row of the alternating sign matrix.

St000067 The inversion number of the alternating sign matrix.

St000076 The rank of the alternating sign matrix in the alternating sign matrix poset.

St000134 The size of the orbit of an alternating sign matrix under gyration.

St000187 The determinant of an alternating sign matrix.

St000193 The row of the unique '1' in the first column of the alternating sign matrix.

St000197 The number of entries equal to positive one in the alternating sign matrix.

St000199 The column of the unique '1' in the last row of the alternating sign matrix.

St000200 The row of the unique '1' in the last column of the alternating sign matrix.

St000227 The osculating paths major index of an alternating sign matrix.

St000332 The positive inversions of an alternating sign matrix.

St000888 The maximal sum of entries on a diagonal of an alternating sign matrix.

St000889 The number of alternating sign matrices with the same antidiagonal sums.

St000890 The number of nonzero entries in an alternating sign matrix.

St000892 The maximal number of nonzero entries on a diagonal of an alternating sign matrix....

St000893 The number of distinct diagonal sums of an alternating sign matrix.

St000894 The trace of an alternating sign matrix.

St000895 The number of ones on the main diagonal of an alternating sign matrix.

St000896 The number of zeros on the main diagonal of an alternating sign matrix.

St000898 The number of maximal entries in the last diagonal of the monotone triangle.

St000066 The column of the unique '1' in the first row of the alternating sign matrix.

St000067 The inversion number of the alternating sign matrix.

St000076 The rank of the alternating sign matrix in the alternating sign matrix poset.

St000134 The size of the orbit of an alternating sign matrix under gyration.

St000187 The determinant of an alternating sign matrix.

St000193 The row of the unique '1' in the first column of the alternating sign matrix.

St000197 The number of entries equal to positive one in the alternating sign matrix.

St000199 The column of the unique '1' in the last row of the alternating sign matrix.

St000200 The row of the unique '1' in the last column of the alternating sign matrix.

St000227 The osculating paths major index of an alternating sign matrix.

St000332 The positive inversions of an alternating sign matrix.

St000888 The maximal sum of entries on a diagonal of an alternating sign matrix.

St000889 The number of alternating sign matrices with the same antidiagonal sums.

St000890 The number of nonzero entries in an alternating sign matrix.

St000892 The maximal number of nonzero entries on a diagonal of an alternating sign matrix....

St000893 The number of distinct diagonal sums of an alternating sign matrix.

St000894 The trace of an alternating sign matrix.

St000895 The number of ones on the main diagonal of an alternating sign matrix.

St000896 The number of zeros on the main diagonal of an alternating sign matrix.

St000898 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
The number of linear extensions of a binary tree.

St000050 The depth/height of a binary tree.

St000051 The size of the left subtree of a binary tree.

St000061 The number of nodes on the left branch of a binary tree.

St000082 The number of elements smaller than a binary tree in Tamari order.

St000083 The number of left oriented leafs of a binary tree except the first one.

St000118 The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary ....

St000121 The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a bin....

St000122 The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a bin....

St000125 The number of occurrences of the contiguous pattern [.,[[[.,.],.],.

St000126 The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a....

St000127 The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a....

St000128 The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a....

St000129 The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a....

St000130 The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a....

St000131 The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

St000132 The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a....

St000161 The sum of the sizes of the right subtrees of a binary tree.

St000196 The number of occurrences of the contiguous pattern [[.,.],[.,.

St000198 A decimal representation of a binary tree as a code word.

St000201 The number of leaf nodes in a binary tree.

St000203 The number of external nodes of a binary tree.

St000204 The number of internal nodes of a binary tree.

St000252 The number of nodes of degree 3 of a binary tree.

St000385 The number of vertices with out-degree 1 in a binary tree.

St000396 The register function (or Horton-Strahler number) of a binary tree.

St000398 The sum of the depths of the vertices (or total internal path length) of a binary....

St000399 The external path length of a binary tree.

St000409 The number of pitchforks in a binary tree.

St000411 The tree factorial of a binary tree.

St000412 The number of binary trees with the same underlying unordered tree.

St000414 The binary logarithm of the number of binary trees with the same underlying unord....

St000568 The hook number of a binary tree.

St000569 The sum of the heights of the vertices of a binary tree.

St000701 The protection number of a binary tree.

St000919 The number of maximal left branches of a binary tree.

St000050 The depth/height of a binary tree.

St000051 The size of the left subtree of a binary tree.

St000061 The number of nodes on the left branch of a binary tree.

St000082 The number of elements smaller than a binary tree in Tamari order.

St000083 The number of left oriented leafs of a binary tree except the first one.

St000118 The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary ....

St000121 The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a bin....

St000122 The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a bin....

St000125 The number of occurrences of the contiguous pattern [.,[[[.,.],.],.

St000126 The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a....

St000127 The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a....

St000128 The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a....

St000129 The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a....

St000130 The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a....

St000131 The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

St000132 The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a....

St000161 The sum of the sizes of the right subtrees of a binary tree.

St000196 The number of occurrences of the contiguous pattern [[.,.],[.,.

St000198 A decimal representation of a binary tree as a code word.

St000201 The number of leaf nodes in a binary tree.

St000203 The number of external nodes of a binary tree.

St000204 The number of internal nodes of a binary tree.

St000252 The number of nodes of degree 3 of a binary tree.

St000385 The number of vertices with out-degree 1 in a binary tree.

St000396 The register function (or Horton-Strahler number) of a binary tree.

St000398 The sum of the depths of the vertices (or total internal path length) of a binary....

St000399 The external path length of a binary tree.

St000409 The number of pitchforks in a binary tree.

St000411 The tree factorial of a binary tree.

St000412 The number of binary trees with the same underlying unordered tree.

St000414 The binary logarithm of the number of binary trees with the same underlying unord....

St000568 The hook number of a binary tree.

St000569 The sum of the heights of the vertices of a binary tree.

St000701 The protection number of a binary tree.

St000919 The number of maximal left branches of a binary tree.

**Binary words**(42 statistics) # word like objects # path like objects

St000288
The number of ones in a binary word.

St000289 The decimal representation of a binary word.

St000290 The major index of a binary word.

St000291 The number of descents of a binary word.

St000292 The number of ascents of a binary word.

St000293 The number of inversions of a binary word.

St000294 The number of distinct factors of a binary word.

St000295 The length of the border of a binary word.

St000296 The length of the symmetric border of a binary word.

St000297 The number of leading ones in a binary word.

St000326 The position of the first one in a non-zero binary word.

St000347 The inversion sum of a binary word.

St000348 The non-inversion sum of a binary word.

St000389 The number of runs of ones of odd length in a binary word.

St000390 The number of runs of ones in a binary word.

St000391 The sum of the positions of the ones in a binary word.

St000392 The length of the longest run of ones in a binary word.

St000393 The number of strictly increasing runs in a binary word.

St000518 The number of distinct subsequences in a binary word.

St000519 The largest length of a factor maximising the subword complexity.

St000529 The number of permutations whose descent word is the given binary word.

St000543 The size of the conjugacy class of a binary word.

St000626 The minimal period of a binary word.

St000627 The exponent of a binary word.

St000628 The balance of a binary word.

St000629 The defect of a binary word.

St000630 The length of the shortest palindromic decomposition of a binary word.

St000631 The number of distinct palindromic decompositions of a binary word.

St000682 The Grundy value of Welter's game on a binary word.

St000691 The number of changes of a binary word.

St000753 The Grundy value for the game of Kayles on a binary word.

St000792 The Grundy value for the game of ruler on a binary word.

St000826 The stopping time of the decimal representation of the binary word for the 3x+1 p....

St000827 The decimal representation of a binary word with a leading 1.

St000847 The number of standard Young tableaux whose descent set is the binary word.

St000875 The semilength of the longest Dyck word in the Catalan factorisation of a binary ....

St000876 The number of factors in the Catalan decomposition of a binary word.

St000877 The depth of the binary word interpreted as a path.

St000878 The number of ones minus the number of zeros of a binary word.

St000885 The number of critical steps in the Catalan decomposition of a binary word.

St000921 The number of internal inversions of a binary word.

St000922 The minimal number such that all substrings of this length are unique.

St000289 The decimal representation of a binary word.

St000290 The major index of a binary word.

St000291 The number of descents of a binary word.

St000292 The number of ascents of a binary word.

St000293 The number of inversions of a binary word.

St000294 The number of distinct factors of a binary word.

St000295 The length of the border of a binary word.

St000296 The length of the symmetric border of a binary word.

St000297 The number of leading ones in a binary word.

St000326 The position of the first one in a non-zero binary word.

St000347 The inversion sum of a binary word.

St000348 The non-inversion sum of a binary word.

St000389 The number of runs of ones of odd length in a binary word.

St000390 The number of runs of ones in a binary word.

St000391 The sum of the positions of the ones in a binary word.

St000392 The length of the longest run of ones in a binary word.

St000393 The number of strictly increasing runs in a binary word.

St000518 The number of distinct subsequences in a binary word.

St000519 The largest length of a factor maximising the subword complexity.

St000529 The number of permutations whose descent word is the given binary word.

St000543 The size of the conjugacy class of a binary word.

St000626 The minimal period of a binary word.

St000627 The exponent of a binary word.

St000628 The balance of a binary word.

St000629 The defect of a binary word.

St000630 The length of the shortest palindromic decomposition of a binary word.

St000631 The number of distinct palindromic decompositions of a binary word.

St000682 The Grundy value of Welter's game on a binary word.

St000691 The number of changes of a binary word.

St000753 The Grundy value for the game of Kayles on a binary word.

St000792 The Grundy value for the game of ruler on a binary word.

St000826 The stopping time of the decimal representation of the binary word for the 3x+1 p....

St000827 The decimal representation of a binary word with a leading 1.

St000847 The number of standard Young tableaux whose descent set is the binary word.

St000875 The semilength of the longest Dyck word in the Catalan factorisation of a binary ....

St000876 The number of factors in the Catalan decomposition of a binary word.

St000877 The depth of the binary word interpreted as a path.

St000878 The number of ones minus the number of zeros of a binary word.

St000885 The number of critical steps in the Catalan decomposition of a binary word.

St000921 The number of internal inversions of a binary word.

St000922 The minimal number such that all substrings of this length are unique.

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

St000158
The length of a core.

St000190 The size of a core.

St000191 The number of strong covers of a core.

St000192 Number of covers of a core in weak Bruhat order.

St000202 The number of $k$-cores contained by a $k$-core.

St000190 The size of a core.

St000191 The number of strong covers of a core.

St000192 Number of covers of a core in weak Bruhat order.

St000202 The number of $k$-cores contained by a $k$-core.

**Dyck paths**(67 statistics) # Catalan objects # path like objects

St000005
The bounce statistic of a Dyck path.

St000006 The dinv statistic of a Dyck path.

St000011 The number of touch points of a Dyck path.

St000012 The area of a Dyck path.

St000013 The height of a Dyck path.

St000014 The number of parking functions supported by a Dyck path.

St000015 The number of peaks of a Dyck path.

St000024 The number of double rises of a Dyck path.

St000025 The number of initial rises of a Dyck path.

St000026 The position of the first return of a Dyck path.

St000027 The major index of a Dyck path.

St000032 The number of elements smaller than the given Dyck path in the Tamari Order.

St000038 The product of the heights of the descending steps of a Dyck path.

St000052 The number of valleys of a Dyck path not on the x-axis.

St000053 The number of valleys of the Dyck path.

St000079 The number of alternating sign matrices for a given Dyck path.

St000117 The number of centered tunnels of a Dyck path.

St000120 The number of left tunnels of a Dyck path.

St000144 The pyramid weight of the Dyck path.

St000306 The bounce count of a Dyck path.

St000329 The number of evenly positioned ascents, with the initial position equal to 1.

St000331 The number of upper interactions of a Dyck path.

St000335 The difference of lower and upper interactions.

St000340 The number of non-final maximal sub-paths of length greater than one.

St000369 The dinv deficit of a Dyck path.

St000376 The bounce deficit of a Dyck path.

St000386 The number of factors DDU in a Dyck path.

St000394 The sum of the heights of the peaks of a Dyck path minus the number of peaks.

St000395 The sum of the heights of the peaks of a Dyck path.

St000418 The number of Dyck paths that are weakly below a Dyck path.

St000419 The number of Dyck paths that are weakly above a Dyck path, except for the path i....

St000420 The number of Dyck paths that are weakly above a Dyck path.

St000421 The number of Dyck paths that are weakly below a Dyck path, except for the path i....

St000438 The position of the last up step in a Dyck path.

St000439 The position of the first down step of a Dyck path.

St000442 The maximal area to the right of an up step of a Dyck path.

St000443 The number of long tunnels of a Dyck path.

St000444 The length of the maximal rise of a Dyck path.

St000445 The number of rises of length 1 of a Dyck path.

St000476 The sum of the semi-lengths of tunnels before a valley of a Dyck path.

St000617 The number of global maxima of a Dyck path.

St000645 The sum of the areas of the rectangles formed by two consecutive peaks and the va....

St000655 The length of the minimal rise of a Dyck path.

St000658 The number of rises of length 2 of a Dyck path.

St000659 The number of rises of length at least 2 of a Dyck path.

St000660 The number of rises of length at least 3 of a Dyck path.

St000661 The number of rises of length 3 of a Dyck path.

St000674 The number of hills of a Dyck path.

St000675 The number of centered multitunnels of a Dyck path.

St000676 The number of odd rises of a Dyck path.

St000678 The number of up steps after the last double rise of a Dyck path.

St000683 The number of points below the Dyck path such that the diagonal to the north-east....

St000684 The global dimension of the LNakayama algebra associated to a Dyck path.

St000685 The dominant dimension of the LNakayama algebra associated to a Dyck path.

St000686 The finitistic dominant dimension of a Dyck path.

St000687 The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.

St000688 The global dimension minus the dominant dimension of the LNakayama algebra associ....

St000689 The maximal n such that the minimal generator-cogenerator module in the LNakayama....

St000790 The number of pairs of centered tunnels, one strictly containing the other, of a ....

St000791 The number of pairs of left tunnels, one strictly containing the other, of a Dyck....

St000874 The position of the last double rise in a Dyck path.

St000920 The logarithmic height of a Dyck path.

St000930 The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.....

St000931 The number of occurrences of the pattern UUU in a Dyck path.

St000932 The number of occurrences of the pattern UDU in a Dyck path.

St000946 The sum of the skew hook positions in a Dyck path.

St000947 The major index of a Dyck path according to Zhao and Zhong.

St000006 The dinv statistic of a Dyck path.

St000011 The number of touch points of a Dyck path.

St000012 The area of a Dyck path.

St000013 The height of a Dyck path.

St000014 The number of parking functions supported by a Dyck path.

St000015 The number of peaks of a Dyck path.

St000024 The number of double rises of a Dyck path.

St000025 The number of initial rises of a Dyck path.

St000026 The position of the first return of a Dyck path.

St000027 The major index of a Dyck path.

St000032 The number of elements smaller than the given Dyck path in the Tamari Order.

St000038 The product of the heights of the descending steps of a Dyck path.

St000052 The number of valleys of a Dyck path not on the x-axis.

St000053 The number of valleys of the Dyck path.

St000079 The number of alternating sign matrices for a given Dyck path.

St000117 The number of centered tunnels of a Dyck path.

St000120 The number of left tunnels of a Dyck path.

St000144 The pyramid weight of the Dyck path.

St000306 The bounce count of a Dyck path.

St000329 The number of evenly positioned ascents, with the initial position equal to 1.

St000331 The number of upper interactions of a Dyck path.

St000335 The difference of lower and upper interactions.

St000340 The number of non-final maximal sub-paths of length greater than one.

St000369 The dinv deficit of a Dyck path.

St000376 The bounce deficit of a Dyck path.

St000386 The number of factors DDU in a Dyck path.

St000394 The sum of the heights of the peaks of a Dyck path minus the number of peaks.

St000395 The sum of the heights of the peaks of a Dyck path.

St000418 The number of Dyck paths that are weakly below a Dyck path.

St000419 The number of Dyck paths that are weakly above a Dyck path, except for the path i....

St000420 The number of Dyck paths that are weakly above a Dyck path.

St000421 The number of Dyck paths that are weakly below a Dyck path, except for the path i....

St000438 The position of the last up step in a Dyck path.

St000439 The position of the first down step of a Dyck path.

St000442 The maximal area to the right of an up step of a Dyck path.

St000443 The number of long tunnels of a Dyck path.

St000444 The length of the maximal rise of a Dyck path.

St000445 The number of rises of length 1 of a Dyck path.

St000476 The sum of the semi-lengths of tunnels before a valley of a Dyck path.

St000617 The number of global maxima of a Dyck path.

St000645 The sum of the areas of the rectangles formed by two consecutive peaks and the va....

St000655 The length of the minimal rise of a Dyck path.

St000658 The number of rises of length 2 of a Dyck path.

St000659 The number of rises of length at least 2 of a Dyck path.

St000660 The number of rises of length at least 3 of a Dyck path.

St000661 The number of rises of length 3 of a Dyck path.

St000674 The number of hills of a Dyck path.

St000675 The number of centered multitunnels of a Dyck path.

St000676 The number of odd rises of a Dyck path.

St000678 The number of up steps after the last double rise of a Dyck path.

St000683 The number of points below the Dyck path such that the diagonal to the north-east....

St000684 The global dimension of the LNakayama algebra associated to a Dyck path.

St000685 The dominant dimension of the LNakayama algebra associated to a Dyck path.

St000686 The finitistic dominant dimension of a Dyck path.

St000687 The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.

St000688 The global dimension minus the dominant dimension of the LNakayama algebra associ....

St000689 The maximal n such that the minimal generator-cogenerator module in the LNakayama....

St000790 The number of pairs of centered tunnels, one strictly containing the other, of a ....

St000791 The number of pairs of left tunnels, one strictly containing the other, of a Dyck....

St000874 The position of the last double rise in a Dyck path.

St000920 The logarithmic height of a Dyck path.

St000930 The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.....

St000931 The number of occurrences of the pattern UUU in a Dyck path.

St000932 The number of occurrences of the pattern UDU in a Dyck path.

St000946 The sum of the skew hook positions in a Dyck path.

St000947 The major index of a Dyck path according to Zhao and Zhong.

**Finite Cartan types**(19 statistics) # Cartan type objects

St000106
The size of the associated Weyl group.

St000107 The dimension of the representation $V(\Lambda_1)$.

St000113 The rank of the Cartan type.

St000138 The Catalan number of an irreducible finite Cartan type.

St000139 The Coxeter number of a finite Cartan type.

St000140 The positive Catalan number of an irreducible finite Cartan type.

St000821 The determinant of the Cartan matrix.

St000851 The third Fuss-Catalan number of a finite Cartan type.

St000852 The second Fuss-Catalan number of a finite Cartan type.

St000853 The number of almost positive roots of a finite Cartan type.

St000854 The number of orbits of reflections of a finite Cartan type.

St000855 The number of full-support reflections in the Weyl group of a finite Cartan type.....

St000856 The number of conjugacy classes in the Weyl group of a finite Cartan type.

St000857 The number of reflections of the Weyl group of a finite Cartan type.

St000858 The number of factorizations of any Coxeter element into reflections of a finite ....

St000859 The number of parking functions of a finite Cartan type.

St000860 The size of the center of the Weyl group of a finite Cartan type.

St000861 The maximal dimension of an irreducible representation of the Weyl group of a fin....

St000865 The number of Coxeter elements in the Weyl group of a finite Cartan type.

St000107 The dimension of the representation $V(\Lambda_1)$.

St000113 The rank of the Cartan type.

St000138 The Catalan number of an irreducible finite Cartan type.

St000139 The Coxeter number of a finite Cartan type.

St000140 The positive Catalan number of an irreducible finite Cartan type.

St000821 The determinant of the Cartan matrix.

St000851 The third Fuss-Catalan number of a finite Cartan type.

St000852 The second Fuss-Catalan number of a finite Cartan type.

St000853 The number of almost positive roots of a finite Cartan type.

St000854 The number of orbits of reflections of a finite Cartan type.

St000855 The number of full-support reflections in the Weyl group of a finite Cartan type.....

St000856 The number of conjugacy classes in the Weyl group of a finite Cartan type.

St000857 The number of reflections of the Weyl group of a finite Cartan type.

St000858 The number of factorizations of any Coxeter element into reflections of a finite ....

St000859 The number of parking functions of a finite Cartan type.

St000860 The size of the center of the Weyl group of a finite Cartan type.

St000861 The maximal dimension of an irreducible representation of the Weyl group of a fin....

St000865 The number of Coxeter elements in the Weyl group of a finite Cartan type.

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

St000072
The number of circled entries.

St000073 The number of boxed entries.

St000074 The number of special entries.

St000077 The number of boxed and circled entries.

St000114 The sum of the entries of the Gelfand-Tsetlin pattern.

St000115 The single entry in the last row.

St000152 The number of boxed plus the number of special entries.

St000176 The total number of tiles in the Gelfand-Tsetlin pattern.

St000177 The number of free tiles in the pattern.

St000178 Number of free entries.

St000186 The sum of the first row in a Gelfand-Tsetlin pattern.

St000073 The number of boxed entries.

St000074 The number of special entries.

St000077 The number of boxed and circled entries.

St000114 The sum of the entries of the Gelfand-Tsetlin pattern.

St000115 The single entry in the last row.

St000152 The number of boxed plus the number of special entries.

St000176 The total number of tiles in the Gelfand-Tsetlin pattern.

St000177 The number of free tiles in the pattern.

St000178 Number of free entries.

St000186 The sum of the first row in a Gelfand-Tsetlin pattern.

**Graphs**(110 statistics) # graph like objects

St000081
The number of edges of a graph.

St000086 The number of subgraphs.

St000087 The number of induced subgraphs.

St000093 The length of the maximal independent set of vertices of a graph.

St000095 The number of triangles of a graph.

St000096 The number of spanning trees of a graph.

St000097 The order of the largest clique of the graph.

St000098 The chromatic number of a graph.

St000171 The degree of the graph.

St000172 The Grundy number of a graph.

St000244 The cardinality of the automorphism group of a graph.

St000258 The burning number of a graph.

St000259 The diameter of a connected graph.

St000260 The radius of a connected graph.

St000261 The edge connectivity of a graph.

St000262 The vertex connectivity of a graph.

St000263 The Szeged index of a graph.

St000264 The girth of a graph, which is not a tree.

St000265 The Wiener index of a graph.

St000266 The number of spanning subgraphs of a graph with the same connected components.

St000267 The number of maximal spanning forests contained in a graph.

St000268 The number of strongly connected orientations of a graph.

St000269 The number of acyclic orientations of a graph.

St000270 The number of forests contained in a graph.

St000271 The chromatic index of a connected graph.

St000272 The treewidth of a graph.

St000273 The domination number of a graph.

St000274 The number of perfect matchings of a graph.

St000276 The size of the preimage of the map 'to graph' from Ordered trees to Graphs.

St000283 The size of the preimage of the map 'to graph' from Binary trees to Graphs.

St000286 The number of connected components of the complement of a graph.

St000287 The number of connected components of a graph.

St000299 The number of nonisomorphic vertex-induced subtrees.

St000300 The number of independent sets of vertices of a graph.

St000301 The number of facets of the stable set polytope of a graph.

St000302 The determinant of the distance matrix of a connected graph.

St000303 The determinant of the product of the incidence matrix and its transpose of a gra....

St000309 The number of vertices with even degree.

St000310 The minimal degree of a vertex of a graph.

St000311 The number of vertices of odd degree in a graph.

St000312 The number of leaves in a graph.

St000313 The number of degree 2 vertices of a graph.

St000315 The number of isolated vertices of a graph.

St000322 The skewness of a graph.

St000323 The minimal crossing number of a graph.

St000343 The number of spanning subgraphs of a graph.

St000344 The number of strongly connected outdegree sequences of a graph.

St000349 The number of different adjacency matrices of a graph.

St000350 The sum of the vertex degrees of a graph.

St000351 The determinant of the adjacency matrix of a graph.

St000361 The second Zagreb index of a graph.

St000362 The size of a minimal vertex cover of a graph.

St000363 The number of minimal vertex covers of a graph.

St000364 The exponent of the automorphism group of a graph.

St000368 The Altshuler-Steinberg determinant of a graph.

St000370 The genus of a graph.

St000379 The number of Hamiltonian cycles in a graph.

St000387 The matching number of a graph.

St000388 The number of orbits of vertices of a graph under automorphisms.

St000403 The Szeged index minus the Wiener index of a graph.

St000422 The energy of a graph, if it is integral.

St000447 The number of pairs of vertices of a graph with distance 3.

St000448 The number of pairs of vertices of a graph with distance 2.

St000449 The number of pairs of vertices of a graph with distance 4.

St000450 The number of edges minus the number of vertices plus 2 of a graph.

St000452 The number of distinct eigenvalues of a graph.

St000453 The number of distinct Laplacian eigenvalues of a graph.

St000454 The largest eigenvalue of a graph if it is integral.

St000455 The second largest eigenvalue of a graph if it is integral.

St000456 The monochromatic index of a connected graph.

St000464 The Schultz index of a connected graph.

St000465 The first Zagreb index of a graph.

St000466 The Gutman (or modified Schultz) index of a connected graph.

St000467 The hyper-Wiener index of a connected graph.

St000468 The Hosoya index of a graph.

St000469 The distinguishing number of a graph.

St000479 The Ramsey number of a graph.

St000482 The (zero)-forcing number of a graph.

St000535 The rank-width of a graph.

St000536 The pathwidth of a graph.

St000537 The cutwidth of a graph.

St000544 The cop number of a graph.

St000552 The number of cut vertices of a graph.

St000553 The number of blocks of a connected graph.

St000571 The F-index (or forgotten topological index) of a graph.

St000636 The hull number of a graph.

St000637 The length of the longest cycle in a graph.

St000671 The maximin edge-connectivity for choosing a subgraph.

St000699 The toughness times the least common multiple of 1,.

St000718 The largest Laplacian eigenvalue of a graph if it is integral.

St000722 The number of different neighbourhoods in a graph.

St000723 The maximal cardinality of a set of vertices with the same neighbourhood in a gra....

St000741 The Colin de Verdière graph invariant.

St000771 The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.....

St000772 The multiplicity of the largest distance Laplacian eigenvalue in a connected grap....

St000773 The multiplicity of the largest Laplacian eigenvalue in a graph.

St000774 The maximal multiplicity of a Laplacian eigenvalue in a graph.

St000775 The multiplicity of the largest eigenvalue in a graph.

St000776 The maximal multiplicity of an eigenvalue in a graph.

St000777 The number of distinct eigenvalues of the distance Laplacian of a connected graph....

St000778 The metric dimension of a graph.

St000785 The number of distinct colouring schemes of a graph.

St000786 The maximal number of occurrences of a colour in a proper colouring of a graph.

St000822 The Hadwiger number of the graph.

St000915 The Ore degree of a graph.

St000916 The packing number of a graph.

St000917 The open packing number of a graph.

St000918 The 2-limited packing number of a graph.

St000926 The clique-coclique number of a graph.

St000948 The chromatic discriminant of a graph.

St000086 The number of subgraphs.

St000087 The number of induced subgraphs.

St000093 The length of the maximal independent set of vertices of a graph.

St000095 The number of triangles of a graph.

St000096 The number of spanning trees of a graph.

St000097 The order of the largest clique of the graph.

St000098 The chromatic number of a graph.

St000171 The degree of the graph.

St000172 The Grundy number of a graph.

St000244 The cardinality of the automorphism group of a graph.

St000258 The burning number of a graph.

St000259 The diameter of a connected graph.

St000260 The radius of a connected graph.

St000261 The edge connectivity of a graph.

St000262 The vertex connectivity of a graph.

St000263 The Szeged index of a graph.

St000264 The girth of a graph, which is not a tree.

St000265 The Wiener index of a graph.

St000266 The number of spanning subgraphs of a graph with the same connected components.

St000267 The number of maximal spanning forests contained in a graph.

St000268 The number of strongly connected orientations of a graph.

St000269 The number of acyclic orientations of a graph.

St000270 The number of forests contained in a graph.

St000271 The chromatic index of a connected graph.

St000272 The treewidth of a graph.

St000273 The domination number of a graph.

St000274 The number of perfect matchings of a graph.

St000276 The size of the preimage of the map 'to graph' from Ordered trees to Graphs.

St000283 The size of the preimage of the map 'to graph' from Binary trees to Graphs.

St000286 The number of connected components of the complement of a graph.

St000287 The number of connected components of a graph.

St000299 The number of nonisomorphic vertex-induced subtrees.

St000300 The number of independent sets of vertices of a graph.

St000301 The number of facets of the stable set polytope of a graph.

St000302 The determinant of the distance matrix of a connected graph.

St000303 The determinant of the product of the incidence matrix and its transpose of a gra....

St000309 The number of vertices with even degree.

St000310 The minimal degree of a vertex of a graph.

St000311 The number of vertices of odd degree in a graph.

St000312 The number of leaves in a graph.

St000313 The number of degree 2 vertices of a graph.

St000315 The number of isolated vertices of a graph.

St000322 The skewness of a graph.

St000323 The minimal crossing number of a graph.

St000343 The number of spanning subgraphs of a graph.

St000344 The number of strongly connected outdegree sequences of a graph.

St000349 The number of different adjacency matrices of a graph.

St000350 The sum of the vertex degrees of a graph.

St000351 The determinant of the adjacency matrix of a graph.

St000361 The second Zagreb index of a graph.

St000362 The size of a minimal vertex cover of a graph.

St000363 The number of minimal vertex covers of a graph.

St000364 The exponent of the automorphism group of a graph.

St000368 The Altshuler-Steinberg determinant of a graph.

St000370 The genus of a graph.

St000379 The number of Hamiltonian cycles in a graph.

St000387 The matching number of a graph.

St000388 The number of orbits of vertices of a graph under automorphisms.

St000403 The Szeged index minus the Wiener index of a graph.

St000422 The energy of a graph, if it is integral.

St000447 The number of pairs of vertices of a graph with distance 3.

St000448 The number of pairs of vertices of a graph with distance 2.

St000449 The number of pairs of vertices of a graph with distance 4.

St000450 The number of edges minus the number of vertices plus 2 of a graph.

St000452 The number of distinct eigenvalues of a graph.

St000453 The number of distinct Laplacian eigenvalues of a graph.

St000454 The largest eigenvalue of a graph if it is integral.

St000455 The second largest eigenvalue of a graph if it is integral.

St000456 The monochromatic index of a connected graph.

St000464 The Schultz index of a connected graph.

St000465 The first Zagreb index of a graph.

St000466 The Gutman (or modified Schultz) index of a connected graph.

St000467 The hyper-Wiener index of a connected graph.

St000468 The Hosoya index of a graph.

St000469 The distinguishing number of a graph.

St000479 The Ramsey number of a graph.

St000482 The (zero)-forcing number of a graph.

St000535 The rank-width of a graph.

St000536 The pathwidth of a graph.

St000537 The cutwidth of a graph.

St000544 The cop number of a graph.

St000552 The number of cut vertices of a graph.

St000553 The number of blocks of a connected graph.

St000571 The F-index (or forgotten topological index) of a graph.

St000636 The hull number of a graph.

St000637 The length of the longest cycle in a graph.

St000671 The maximin edge-connectivity for choosing a subgraph.

St000699 The toughness times the least common multiple of 1,.

St000718 The largest Laplacian eigenvalue of a graph if it is integral.

St000722 The number of different neighbourhoods in a graph.

St000723 The maximal cardinality of a set of vertices with the same neighbourhood in a gra....

St000741 The Colin de Verdière graph invariant.

St000771 The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.....

St000772 The multiplicity of the largest distance Laplacian eigenvalue in a connected grap....

St000773 The multiplicity of the largest Laplacian eigenvalue in a graph.

St000774 The maximal multiplicity of a Laplacian eigenvalue in a graph.

St000775 The multiplicity of the largest eigenvalue in a graph.

St000776 The maximal multiplicity of an eigenvalue in a graph.

St000777 The number of distinct eigenvalues of the distance Laplacian of a connected graph....

St000778 The metric dimension of a graph.

St000785 The number of distinct colouring schemes of a graph.

St000786 The maximal number of occurrences of a colour in a proper colouring of a graph.

St000822 The Hadwiger number of the graph.

St000915 The Ore degree of a graph.

St000916 The packing number of a graph.

St000917 The open packing number of a graph.

St000918 The 2-limited packing number of a graph.

St000926 The clique-coclique number of a graph.

St000948 The chromatic discriminant of a graph.

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

St000008
The major index of the composition.

St000047 The number of standard immaculate tableaux of a given shape.

St000089 The absolute variation of a composition.

St000090 The variation of a composition.

St000091 The descent variation of a composition.

St000277 The size of the preimage of the map 'descent composition' from Permutations to In....

St000285 The size of the preimage of the map 'to inverse des composition' from Parking fun....

St000381 The largest part of an integer composition.

St000382 The first part of an integer composition.

St000383 The last part of an integer composition.

St000657 The smallest part of an integer composition.

St000757 The length of the longest weakly inreasing subsequence of parts of an integer com....

St000758 The length of the longest staircase fitting into an integer composition.

St000760 The length of the longest strictly decreasing subsequence of parts of an integer ....

St000761 The number of ascents in an integer composition.

St000762 The sum of the positions of the weak records of an integer composition.

St000763 The sum of the positions of the strong records of an integer composition.

St000764 The number of strong records in an integer composition.

St000765 The number of weak records in an integer composition.

St000766 The number of inversions of an integer composition.

St000767 The number of runs in an integer composition.

St000768 The number of peaks in an integer composition.

St000769 The major index of a composition.

St000805 The number of peaks of the associated bargraph.

St000806 The semiperimeter of the associated bargraph.

St000807 The sum of the heights of the valleys of the associated bargraph.

St000808 The number of up steps of the associated bargraph.

St000816 The number of standard composition tableaux of the composition.

St000817 The sum of the entries in the column specified by the composition of the change o....

St000818 The sum of the entries in the column specified by the composition of the change o....

St000820 The number of compositions obtained by rotating the composition.

St000899 The maximal number of repetitions of an integer composition.

St000900 The minimal number of repetitions of a part in an integer composition.

St000902 The minimal number of repetitions of an integer composition.

St000903 The number of different parts of an integer composition.

St000904 The maximal number of repetitions of an integer composition.

St000905 The number of different multiplicities of parts of an integer composition.

St000047 The number of standard immaculate tableaux of a given shape.

St000089 The absolute variation of a composition.

St000090 The variation of a composition.

St000091 The descent variation of a composition.

St000277 The size of the preimage of the map 'descent composition' from Permutations to In....

St000285 The size of the preimage of the map 'to inverse des composition' from Parking fun....

St000381 The largest part of an integer composition.

St000382 The first part of an integer composition.

St000383 The last part of an integer composition.

St000657 The smallest part of an integer composition.

St000757 The length of the longest weakly inreasing subsequence of parts of an integer com....

St000758 The length of the longest staircase fitting into an integer composition.

St000760 The length of the longest strictly decreasing subsequence of parts of an integer ....

St000761 The number of ascents in an integer composition.

St000762 The sum of the positions of the weak records of an integer composition.

St000763 The sum of the positions of the strong records of an integer composition.

St000764 The number of strong records in an integer composition.

St000765 The number of weak records in an integer composition.

St000766 The number of inversions of an integer composition.

St000767 The number of runs in an integer composition.

St000768 The number of peaks in an integer composition.

St000769 The major index of a composition.

St000805 The number of peaks of the associated bargraph.

St000806 The semiperimeter of the associated bargraph.

St000807 The sum of the heights of the valleys of the associated bargraph.

St000808 The number of up steps of the associated bargraph.

St000816 The number of standard composition tableaux of the composition.

St000817 The sum of the entries in the column specified by the composition of the change o....

St000818 The sum of the entries in the column specified by the composition of the change o....

St000820 The number of compositions obtained by rotating the composition.

St000899 The maximal number of repetitions of an integer composition.

St000900 The minimal number of repetitions of a part in an integer composition.

St000902 The minimal number of repetitions of an integer composition.

St000903 The number of different parts of an integer composition.

St000904 The maximal number of repetitions of an integer composition.

St000905 The number of different multiplicities of parts of an integer composition.

**Integer partitions**(126 statistics) # partition like objects

St000003
The number of standard Young tableaux of the partition.

St000010 The length of the partition.

St000046 The largest eigenvalue of the random to random operator acting on the simple modu....

St000048 The multinomial of the parts of a partition.

St000049 The number of set partitions whose sorted block sizes correspond to the partition....

St000063 The number of linear extensions of a certain poset defined from a partition \la....

St000088 The row sums of the character table of the symmetric group.

St000108 The number of partitions contained in the given partition.

St000137 The Grundy value of an integer partition.

St000142 The number of even parts of a partition.

St000143 The maximal repeated part of a partition.

St000145 The Dyson rank of a partition.

St000146 The Andrews-Garvan crank of a partition.

St000147 The largest part of an integer partition.

St000148 The number of odd parts of a partition.

St000149 The number of cells of the partition whose leg is zero and arm is odd.

St000150 The floored half-sum of the multiplicities of a partition.

St000159 The number of different parts of an integer partition.

St000160 Multiplicity of the smallest part of $\lambda$.

St000175 Degree of the polynomial counting the number of semistandard Young-tableaux of sh....

St000179 Product of hook lengths.

St000182 The number of permutations whose cycle type is the given integer partition.

St000183 The size of the Durfee Square of a partition.

St000184 The size of the centralizer of any permutation of given cycle type.

St000185 The weighted size of a partition.

St000205 Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and part....

St000206 Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and inte....

St000207 Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

St000208 Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

St000212 Number of standard Young tableaux for an integer partition such that no two conse....

St000225 Difference between largest and smallest parts in a partition.

St000228 The size of a partition.

St000256 The number of parts from which one can substract 2 and still get an integer parti....

St000257 The number of distinct parts of a partition that occur at least twice.

St000275 Number of permutations whose sorted list of non zero multiplicities of the Lehmer....

St000278 The size of the preimage of the map 'to partition' from Integer compositions to I....

St000284 The Plancherel distribution on integer partitions.

St000318 The number of addable cells of the Ferrers diagram of an integer partition.

St000319 The spin of an integer partition.

St000320 The dinv adjustment of an integer partition.

St000321 The number of integer partitions of n that are dominated by an integer partition.....

St000345 The number of refinements of a partition.

St000346 The number of coarsenings of a partition.

St000377 The dinv defect of an integer partition.

St000378 The diagonal inversion number of an integer partition.

St000380 Half the perimeter of the largest rectangle that fits inside the diagram of an in....

St000384 The maximal part of the shifted composition of an integer partition.

St000459 The hook length of the base cell of a partition.

St000460 The hook length of the last cell along the main diagonal of an integer partition.....

St000473 The number of parts of a partition that are strictly bigger than the number of on....

St000474 Dyson's crank of a partition.

St000475 The number of parts equal to 1 in a partition.

St000477 The weight of a partition according to Alladi.

St000478 Another weight of a partition according to Alladi.

St000480 The number of lower covers of a partition in dominance order.

St000481 The number of upper covers of a partition in dominance order.

St000506 The number of standard desarrangement tableaux of shape equal to the given partit....

St000509 The diagonal index (content) of a partition.

St000510 The number of invariant oriented cycles when acting with a permutation of given c....

St000511 The number of invariant subsets when acting with a permutation of given cycle typ....

St000512 The number of invariant subsets of size 3 when acting with a permutation of given....

St000513 The number of invariant subsets of size 2 when acting with a permutation of given....

St000514 The number of invariant simple graphs when acting with a permutation of given cyc....

St000515 The number of invariant set partitions when acting with a permutation of given cy....

St000517 The Kreweras number of an integer partition.

St000531 The number of ways to place as many non-attacking rooks as possible on a Ferrers ....

St000532 The total number of rook placements on a Ferrers board.

St000533 The maximal number of non-attacking rooks on a Ferrers shape.

St000547 The number of even non-empty partial sums of an integer partition.

St000548 The number of different non-empty partial sums of an integer partition.

St000549 The number of odd partial sums of an integer partition.

St000566 The number of ways to select a row of a Ferrers shape and two cells in this row.

St000567 The sum of the products of all pairs of parts.

St000618 The number of self-evacuating tableaux of given shape.

St000620 The number of standard tableaux of shape equal to the given partition such that t....

St000621 The number of standard tableaux of shape equal to the given partition such that t....

St000644 The number of graphs with given frequency partition.

St000667 The greatest common divisor of the parts of the partition.

St000668 The least common multiple of the parts of the partition.

St000681 The Grundy value of Chomp on Ferrers diagrams.

St000697 The number of 3-rim hooks removed from an integer partition to obtain its associa....

St000698 The number of 2-rim hooks removed from an integer partition to obtain its associa....

St000704 The number of semistandard tableaux on a given integer partition with minimal max....

St000705 The number of semistandard tableaux on a given integer partition of n with maxima....

St000706 The product of the factorials of the multiplicities of an integer partition.

St000707 The product of the factorials of the parts.

St000708 The product of the parts of an integer partition.

St000712 The number of semistandard Young tableau of given shape, with entries at most 4.

St000713 The dimension of the irreducible representation of Sp(4) labelled by an integer p....

St000714 The number of semistandard Young tableau of given shape, with entries at most 2.

St000715 The number of semistandard Young tableau of given shape, with entries at most 3.

St000716 The dimension of the irreducible representation of Sp(6) labelled by an integer p....

St000749 The smallest integer d such that the restriction of the representation correspond....

St000752 The Grundy value for the game 'Couples are forever' on an integer partition.

St000755 The number of real roots of the characteristic polynomial of a linear recurrence ....

St000759 The smallest missing part in an integer partition.

St000770 The major index of an integer partition when read from bottom to top.

St000781 The number of proper colouring schemes of a Ferrers diagram.

St000783 The maximal number of occurrences of a colour in a proper colouring of a Ferrers ....

St000784 The maximum of the length and the number of parts of an integer partition.

St000810 The sum of the entries in the column specified by the partition of the change of ....

St000811 The sum of the entries in the column specified by the partition of the change of ....

St000812 The sum of the entries in the column specified by the partition of the change of ....

St000813 The number of zero-one matrices with weakly decreasing column sums and row sums g....

St000814 The sum of the entries in the column specified by the partition of the change of ....

St000815 The number of semistandard Young tableaux of partition weight of given shape.

St000835 The minimal difference in size when partitioning the integer partition into two s....

St000867 The sum of the hook lengths in the first column of an integer partition.

St000869 The sum of the hook lengths of an integer partition.

St000870 The product of the hook lengths of the diagonal cells in an integer partition.

St000897 The number of different multiplicities of parts of an integer partition.

St000901 The cube of the number of standard Young tableaux with shape given by the partiti....

St000913 The number of ways to refine the partition into singletons.

St000927 The alternating sum of the coefficients of the character polynomial of an integer....

St000928 The sum of the coefficients of the character polynomial of an integer partition.

St000929 The constant term of the character polynomial of an integer partition.

St000933 The number of multipartitions of sizes given by an integer partition.

St000934 The 2-degree of an integer partition.

St000935 The number of ordered refinements of an integer partition.

St000936 The number of even values of the symmetric group character corresponding to the p....

St000937 The number of positive values of the symmetric group character corresponding to t....

St000938 The number of zeros of the symmetric group character corresponding to the partiti....

St000939 The number of characters of the symmetric group whose value on the partition is p....

St000940 The number of characters of the symmetric group whose value on the partition is z....

St000941 The number of characters of the symmetric group whose value on the partition is e....

St000944 The 3-degree of an integer partition.

St000010 The length of the partition.

St000046 The largest eigenvalue of the random to random operator acting on the simple modu....

St000048 The multinomial of the parts of a partition.

St000049 The number of set partitions whose sorted block sizes correspond to the partition....

St000063 The number of linear extensions of a certain poset defined from a partition \la....

St000088 The row sums of the character table of the symmetric group.

St000108 The number of partitions contained in the given partition.

St000137 The Grundy value of an integer partition.

St000142 The number of even parts of a partition.

St000143 The maximal repeated part of a partition.

St000145 The Dyson rank of a partition.

St000146 The Andrews-Garvan crank of a partition.

St000147 The largest part of an integer partition.

St000148 The number of odd parts of a partition.

St000149 The number of cells of the partition whose leg is zero and arm is odd.

St000150 The floored half-sum of the multiplicities of a partition.

St000159 The number of different parts of an integer partition.

St000160 Multiplicity of the smallest part of $\lambda$.

St000175 Degree of the polynomial counting the number of semistandard Young-tableaux of sh....

St000179 Product of hook lengths.

St000182 The number of permutations whose cycle type is the given integer partition.

St000183 The size of the Durfee Square of a partition.

St000184 The size of the centralizer of any permutation of given cycle type.

St000185 The weighted size of a partition.

St000205 Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and part....

St000206 Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and inte....

St000207 Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

St000208 Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....

St000212 Number of standard Young tableaux for an integer partition such that no two conse....

St000225 Difference between largest and smallest parts in a partition.

St000228 The size of a partition.

St000256 The number of parts from which one can substract 2 and still get an integer parti....

St000257 The number of distinct parts of a partition that occur at least twice.

St000275 Number of permutations whose sorted list of non zero multiplicities of the Lehmer....

St000278 The size of the preimage of the map 'to partition' from Integer compositions to I....

St000284 The Plancherel distribution on integer partitions.

St000318 The number of addable cells of the Ferrers diagram of an integer partition.

St000319 The spin of an integer partition.

St000320 The dinv adjustment of an integer partition.

St000321 The number of integer partitions of n that are dominated by an integer partition.....

St000345 The number of refinements of a partition.

St000346 The number of coarsenings of a partition.

St000377 The dinv defect of an integer partition.

St000378 The diagonal inversion number of an integer partition.

St000380 Half the perimeter of the largest rectangle that fits inside the diagram of an in....

St000384 The maximal part of the shifted composition of an integer partition.

St000459 The hook length of the base cell of a partition.

St000460 The hook length of the last cell along the main diagonal of an integer partition.....

St000473 The number of parts of a partition that are strictly bigger than the number of on....

St000474 Dyson's crank of a partition.

St000475 The number of parts equal to 1 in a partition.

St000477 The weight of a partition according to Alladi.

St000478 Another weight of a partition according to Alladi.

St000480 The number of lower covers of a partition in dominance order.

St000481 The number of upper covers of a partition in dominance order.

St000506 The number of standard desarrangement tableaux of shape equal to the given partit....

St000509 The diagonal index (content) of a partition.

St000510 The number of invariant oriented cycles when acting with a permutation of given c....

St000511 The number of invariant subsets when acting with a permutation of given cycle typ....

St000512 The number of invariant subsets of size 3 when acting with a permutation of given....

St000513 The number of invariant subsets of size 2 when acting with a permutation of given....

St000514 The number of invariant simple graphs when acting with a permutation of given cyc....

St000515 The number of invariant set partitions when acting with a permutation of given cy....

St000517 The Kreweras number of an integer partition.

St000531 The number of ways to place as many non-attacking rooks as possible on a Ferrers ....

St000532 The total number of rook placements on a Ferrers board.

St000533 The maximal number of non-attacking rooks on a Ferrers shape.

St000547 The number of even non-empty partial sums of an integer partition.

St000548 The number of different non-empty partial sums of an integer partition.

St000549 The number of odd partial sums of an integer partition.

St000566 The number of ways to select a row of a Ferrers shape and two cells in this row.

St000567 The sum of the products of all pairs of parts.

St000618 The number of self-evacuating tableaux of given shape.

St000620 The number of standard tableaux of shape equal to the given partition such that t....

St000621 The number of standard tableaux of shape equal to the given partition such that t....

St000644 The number of graphs with given frequency partition.

St000667 The greatest common divisor of the parts of the partition.

St000668 The least common multiple of the parts of the partition.

St000681 The Grundy value of Chomp on Ferrers diagrams.

St000697 The number of 3-rim hooks removed from an integer partition to obtain its associa....

St000698 The number of 2-rim hooks removed from an integer partition to obtain its associa....

St000704 The number of semistandard tableaux on a given integer partition with minimal max....

St000705 The number of semistandard tableaux on a given integer partition of n with maxima....

St000706 The product of the factorials of the multiplicities of an integer partition.

St000707 The product of the factorials of the parts.

St000708 The product of the parts of an integer partition.

St000712 The number of semistandard Young tableau of given shape, with entries at most 4.

St000713 The dimension of the irreducible representation of Sp(4) labelled by an integer p....

St000714 The number of semistandard Young tableau of given shape, with entries at most 2.

St000715 The number of semistandard Young tableau of given shape, with entries at most 3.

St000716 The dimension of the irreducible representation of Sp(6) labelled by an integer p....

St000749 The smallest integer d such that the restriction of the representation correspond....

St000752 The Grundy value for the game 'Couples are forever' on an integer partition.

St000755 The number of real roots of the characteristic polynomial of a linear recurrence ....

St000759 The smallest missing part in an integer partition.

St000770 The major index of an integer partition when read from bottom to top.

St000781 The number of proper colouring schemes of a Ferrers diagram.

St000783 The maximal number of occurrences of a colour in a proper colouring of a Ferrers ....

St000784 The maximum of the length and the number of parts of an integer partition.

St000810 The sum of the entries in the column specified by the partition of the change of ....

St000811 The sum of the entries in the column specified by the partition of the change of ....

St000812 The sum of the entries in the column specified by the partition of the change of ....

St000813 The number of zero-one matrices with weakly decreasing column sums and row sums g....

St000814 The sum of the entries in the column specified by the partition of the change of ....

St000815 The number of semistandard Young tableaux of partition weight of given shape.

St000835 The minimal difference in size when partitioning the integer partition into two s....

St000867 The sum of the hook lengths in the first column of an integer partition.

St000869 The sum of the hook lengths of an integer partition.

St000870 The product of the hook lengths of the diagonal cells in an integer partition.

St000897 The number of different multiplicities of parts of an integer partition.

St000901 The cube of the number of standard Young tableaux with shape given by the partiti....

St000913 The number of ways to refine the partition into singletons.

St000927 The alternating sum of the coefficients of the character polynomial of an integer....

St000928 The sum of the coefficients of the character polynomial of an integer partition.

St000929 The constant term of the character polynomial of an integer partition.

St000933 The number of multipartitions of sizes given by an integer partition.

St000934 The 2-degree of an integer partition.

St000935 The number of ordered refinements of an integer partition.

St000936 The number of even values of the symmetric group character corresponding to the p....

St000937 The number of positive values of the symmetric group character corresponding to t....

St000938 The number of zeros of the symmetric group character corresponding to the partiti....

St000939 The number of characters of the symmetric group whose value on the partition is p....

St000940 The number of characters of the symmetric group whose value on the partition is z....

St000941 The number of characters of the symmetric group whose value on the partition is e....

St000944 The 3-degree of an integer partition.

**Ordered trees**(19 statistics) # tree like structures # Catalan objects # graph like objects

St000084
The number of subtrees.

St000085 The number of linear extensions of the tree.

St000094 The depth of an ordered tree.

St000166 The depth minus 1 of an ordered tree.

St000167 The number of leaves of an ordered tree.

St000168 The number of internal nodes of an ordered tree.

St000328 The maximum number of child nodes in a tree.

St000397 The Strahler number of a rooted tree.

St000400 The path length of an ordered tree.

St000410 The tree factorial of an ordered tree.

St000413 The number of ordered trees with the same underlying unordered tree.

St000415 The size of the automorphism group of the rooted tree underlying the ordered tree....

St000416 The number of inequivalent increasing trees of an ordered tree.

St000417 The size of the automorphism group of the ordered tree.

St000521 The number of distinct subtrees of an ordered tree.

St000522 The number of 1-protected nodes of a rooted tree.

St000523 The number of 2-protected nodes of a rooted tree.

St000679 The pruning number of an ordered tree.

St000700 The protection number of an ordered tree.

St000085 The number of linear extensions of the tree.

St000094 The depth of an ordered tree.

St000166 The depth minus 1 of an ordered tree.

St000167 The number of leaves of an ordered tree.

St000168 The number of internal nodes of an ordered tree.

St000328 The maximum number of child nodes in a tree.

St000397 The Strahler number of a rooted tree.

St000400 The path length of an ordered tree.

St000410 The tree factorial of an ordered tree.

St000413 The number of ordered trees with the same underlying unordered tree.

St000415 The size of the automorphism group of the rooted tree underlying the ordered tree....

St000416 The number of inequivalent increasing trees of an ordered tree.

St000417 The size of the automorphism group of the ordered tree.

St000521 The number of distinct subtrees of an ordered tree.

St000522 The number of 1-protected nodes of a rooted tree.

St000523 The number of 2-protected nodes of a rooted tree.

St000679 The pruning number of an ordered tree.

St000700 The protection number of an ordered tree.

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

St000135
The number of lucky cars of the parking function.

St000136 The dinv of a parking function.

St000165 Sum of the entries.

St000188 The cosum of a parking function.

St000194 The number of primary dinversion pairs of a labelled dyck path corresponding to a....

St000195 The number of secondary dinversion pairs of the dyck path corresponding to a park....

St000540 The sum of the entries of a parking function minus its length.

St000942 The number of critical left to right maxima of the parking functions.

St000943 The number of spots the most unlucky car had to go further in a parking function.....

St000136 The dinv of a parking function.

St000165 Sum of the entries.

St000188 The cosum of a parking function.

St000194 The number of primary dinversion pairs of a labelled dyck path corresponding to a....

St000195 The number of secondary dinversion pairs of the dyck path corresponding to a park....

St000540 The sum of the entries of a parking function minus its length.

St000942 The number of critical left to right maxima of the parking functions.

St000943 The number of spots the most unlucky car had to go further in a parking function.....

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

St000041
The number of nestings of a perfect matching.

St000042 The number of crossings of a perfect matching.

St000043 The number of crossings plus two-nestings of a perfect matching.

St000044 The number of vertices of the unicellular map given by a perfect matching.

St000164 The number of short pairs.

St000719 The number of alignments in a perfect matching.

St000720 The size of the largest partition in the oscillating tableau corresponding to the....

St000721 The sum of the partition sizes in the oscillating tableau corresponding to a perf....

St000746 The number of pairs with odd minimum in a perfect matching.

St000754 The Grundy value for the game of removing nestings in a perfect matching.

St000780 The size of the orbit under rotation of a perfect matching.

St000782 The indicator function of whether a given perfect matching is an L & P matching.

St000787 The number of flips required to make a perfect matching noncrossing.

St000788 The number of nesting-similar perfect matchings of a perfect matching.

St000789 The number of crossing-similar perfect matchings of a perfect matching.

St000819 The propagating number of a perfect matching.

St000838 The number of terminal right-hand endpoints when the vertices are written in orde....

St000840 The number of closers smaller than the largest opener in a perfect matching.

St000841 The largest opener of a perfect matching.

St000843 The decomposition number of a perfect matching.

St000924 The number of topologically connected components of a perfect matching.

St000945 The number of matchings in the dihedral orbit of a perfect matching.

St000042 The number of crossings of a perfect matching.

St000043 The number of crossings plus two-nestings of a perfect matching.

St000044 The number of vertices of the unicellular map given by a perfect matching.

St000164 The number of short pairs.

St000719 The number of alignments in a perfect matching.

St000720 The size of the largest partition in the oscillating tableau corresponding to the....

St000721 The sum of the partition sizes in the oscillating tableau corresponding to a perf....

St000746 The number of pairs with odd minimum in a perfect matching.

St000754 The Grundy value for the game of removing nestings in a perfect matching.

St000780 The size of the orbit under rotation of a perfect matching.

St000782 The indicator function of whether a given perfect matching is an L & P matching.

St000787 The number of flips required to make a perfect matching noncrossing.

St000788 The number of nesting-similar perfect matchings of a perfect matching.

St000789 The number of crossing-similar perfect matchings of a perfect matching.

St000819 The propagating number of a perfect matching.

St000838 The number of terminal right-hand endpoints when the vertices are written in orde....

St000840 The number of closers smaller than the largest opener in a perfect matching.

St000841 The largest opener of a perfect matching.

St000843 The decomposition number of a perfect matching.

St000924 The number of topologically connected components of a perfect matching.

St000945 The number of matchings in the dihedral orbit of a perfect matching.

**Permutations**(252 statistics) # word like objects

St000001
The number of ways to write a permutation as a minimal length product of simple t....

St000002 The number of occurrences of the pattern 123 in a permutation.

St000004 The major index of a permutation.

St000007 The number of saliances of the permutation.

St000018 The number of inversions of a permutation.

St000019 The cardinality of the complement of the connectivity set.

St000020 The rank of the permutation.

St000021 The number of descents of a permutation.

St000022 The number of fixed points of a permutation.

St000023 The number of inner peaks of a permutation.

St000028 The number of stack-sorts needed to sort a permutation.

St000029 The depth of a permutation.

St000030 The sum of the descent differences of a permutations.

St000031 The number of cycles in the cycle decomposition of a permutation.

St000033 The number of permutations greater than or equal to the given permutation in (str....

St000034 The maximum defect over any reduced expression for a permutation and any subexpre....

St000035 The number of left outer peaks of a permutation.

St000036 The evaluation of the Kazhdan-Lusztig polynomial $P(id,w)$ for each permutation $w$ i....

St000037 The sign of a permutation.

St000039 The number of crossings of a permutation.

St000040 The number of regions of inversion arrangement of a permutation.

St000054 The first entry of the permutation.

St000055 The inversion sum of a permutation.

St000056 The decomposition (or block) number of a permutation.

St000058 The order of a permutation.

St000060 The greater neighbor of the maximum.

St000062 The length of the longest increasing subsequence of the permutation.

St000064 The number of one-box pattern of a permutation.

St000078 The number of alternating sign matrices whose left key is the permutation.

St000092 The number of outer peaks of a permutation.

St000099 The number of valleys of a permutation, including the boundary.

St000109 The number of elements less than or equal to the given element in Bruhat order.

St000110 The number of permutations less than or equal to given permutation in left weak o....

St000111 The sum of the descent tops (or Genocchi descents) of a permutation.

St000119 The number of occurrences of the pattern 321 in a permutation.

St000123 The difference in Coxeter length of a permutation and its image under the Simion-....

St000124 The cardinality of the preimage of the Simion-Schmidt map.

St000133 The "bounce" of a permutation.

St000141 The maximum drop size of a permutation.

St000153 The number of adjacent cycles of a permutation.

St000154 The sum of the descent bottoms of a permutation.

St000155 The number of exceedances (also excedences) of a permutation.

St000156 The Denert index of a permutation.

St000162 The number of nontrivial cycles of a permutation $\pi$ in its cycle decomposition.

St000209 Maximum difference of elements in cycles.

St000210 Minimum over maximum difference of elements in cycles.

St000213 The number of weak exceedances (also weak excedences) of a permutation.

St000214 The number of adjacencies (or small descents) of a permutation.

St000215 The number of adjacencies of a permutation, 0 appended.

St000216 The absolute length of a permutation.

St000217 The number of occurrences of the pattern 312 in a permutation.

St000218 The number of occurrences of the pattern 213 in a permutation.

St000219 The number of occurrences of the pattern 231 in a permutation.

St000220 The number of occurrences of the pattern 132 in a permutation.

St000221 The number of strong fixed points of a permutation.

St000222 The number of alignments of a permutation

St000223 The number of nestings of a permutation

St000224 The sorting index of a permutation.

St000226 The convexity of a permutation.

St000234 The number of global ascents of a permutation.

St000235 The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically.

St000236 The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

St000237 The number of indices $i$ such that $\pi_i=i+1$.

St000238 The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.

St000239 The number of indices $i$ such that $\pi_i \in \{i,i+1\}$.

St000240 The number of indices $i$ for which $\pi_i \neq i+1$.

St000241 The number of indices $i$ such that $\pi_i = i+1$ considered cyclically.

St000242 The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically.....

St000243 The number of cyclic valleys and cyclic peaks of a permutation.

St000245 The number of ascents of a permutation.

St000246 The number of non-inversions of a permutation.

St000255 The number of reduced Kogan faces with the permutation as type.

St000279 The size of the preimage of the map 'cycle-as-one-line notation' from Permutation....

St000280 The size of the preimage of the map 'to labelling permutation' from Parking funct....

St000304 The load of a permutation.

St000305 The inverse major index of a permutation.

St000308 The height of the tree associated to a permutation.

St000314 The number of left-to-right-maxima of a permutation.

St000316 The number of non-left-to-right-maxima of a permutation.

St000317 The cycle descent number of a permutation.

St000324 The shape of the tree associated to a permutation.

St000325 The width of the tree associated to a permutation.

St000333 The dez statistic, the number of descents of a permutation after replacing fixed ....

St000334 The maz index, the major index of a permutation after replacing fixed points by z....

St000337 The lec statistic, the sum of the inversion numbers of the hook factors of a perm....

St000338 The number of pixed points of a permutation.

St000339 The maf index of a permutation.

St000341 The non-inversion sum of a permutation.

St000342 The cosine of a permutation.

St000352 The Elizalde-Pak rank of a permutation.

St000353 The number of inner valleys of a permutation.

St000354 The number of recoils of a permutation.

St000355 The number of occurrences of the pattern 21-3.

St000356 The number of occurrences of the pattern 13-2.

St000357 The number of occurrences of the pattern 12-3.

St000358 The number of occurrences of the pattern 31-2.

St000359 The number of occurrences of the pattern 23-1.

St000360 The number of occurrences of the pattern 32-1.

St000365 The number of double ascents of a permutation.

St000366 The number of double descents of a permutation.

St000367 The number of simsun double descents of a permutation.

St000371 The number of positions mid points of decreasing subsequences of length 3 in a pe....

St000372 The number of positions of mid points of increasing subsequences of length 3 in a....

St000373 The number of weak exceedences of a permutation that are also mid-points of a dec....

St000374 The number of non weak exceedences of a permutation that are also not mid-points ....

St000375 The number of non weak exceedences of a permutation that are mid-points of a decr....

St000401 The size of the symmetry class of a permutation.

St000402 Half the size of the symmetry class of a permutation.

St000404 The number of occurrences of the pattern 3241 or of the pattern 4231 in a permuta....

St000405 The number of occurrences of the pattern 1324 in a permutation.

St000406 The number of occurrences of the pattern 3241 in a permutation.

St000407 The number of occurrences of the pattern 2143 in a permutation.

St000408 The number of occurrences of the pattern 4231 in a permutation.

St000423 The number of occurrences of the pattern 123 or of the pattern 132 in a permutati....

St000424 The number of occurrences of the pattern 132 or of the pattern 231 in a permutati....

St000425 The number of occurrences of the pattern 132 or of the pattern 213 in a permutati....

St000426 The number of occurrences of the pattern 132 or of the pattern 312 in a permutati....

St000427 The number of occurrences of the pattern 123 or of the pattern 231 in a permutati....

St000428 The number of occurrences of the pattern 123 or of the pattern 213 in a permutati....

St000429 The number of occurrences of the pattern 123 or of the pattern 321 in a permutati....

St000430 The number of occurrences of the pattern 123 or of the pattern 312 in a permutati....

St000431 The number of occurrences of the pattern 213 or of the pattern 321 in a permutati....

St000432 The number of occurrences of the pattern 231 or of the pattern 312 in a permutati....

St000433 The number of occurrences of the pattern 132 or of the pattern 321 in a permutati....

St000434 The number of occurrences of the pattern 213 or of the pattern 312 in a permutati....

St000435 The number of occurrences of the pattern 213 or of the pattern 231 in a permutati....

St000436 The number of occurrences of the pattern 231 or of the pattern 321 in a permutati....

St000437 The number of occurrences of the pattern 312 or of the pattern 321 in a permutati....

St000440 The number of occurrences of the pattern 4132 or of the pattern 4231 in a permuta....

St000441 The number of successions (or small ascents) of a permutation.

St000446 The disorder of a permutation.

St000451 The length of the longest pattern of the form k 1 2.

St000457 The number of occurrences of one of the patterns 132, 213 or 321 in a permutation....

St000458 The number of permutations obtained by switching adjacencies or successions.

St000461 The rix statistic of a permutation.

St000462 The major index minus the number of excedences of a permutation.

St000463 The number of admissible inversions of a permutation.

St000470 The number of runs in a permutation.

St000471 The sum of the ascent tops of a permutation.

St000472 The sum of the ascent bottoms of a permutation.

St000483 The number of times a permutation switches from increasing to decreasing or decre....

St000484 The sum of St000483 over all subsequences of length at least three.

St000485 The length of the longest cycle of a permutation.

St000486 The number of cycles of length at least 3 of a permutation.

St000487 The length of the shortest cycle of a permutation.

St000488 The number of cycle of a permutation of length at most 2.

St000489 The number of cycle of a permutation of length at most 3.

St000494 The number of inversions of distance at most 3 of a permutation.

St000495 The number of inversions of distance at most 2 of a permutation.

St000500 Eigenvalues of the random-to-random operator acting on the regular representation....

St000501 The size of the first part in the decomposition of a permutation.

St000516 The number of stretching pairs of a permutation.

St000520 The number of subsequences of a permutation which are not order-isomorphic.

St000530 The number of permutations with the same descent word as the given permutation.

St000534 The number of 2-rises of a permutation.

St000538 The number of even inversions of a permutation.

St000539 The number of odd inversions of a permutation.

St000541 The number of indices greater than or equal to 2 of a permutation such that all s....

St000542 The number of left-to-right-minima of a permutation.

St000545 The number of parabolic double cosets with minimal element being the given permut....

St000546 The number of global descents of a permutation.

St000570 The Edelman-Greene number of a permutation.

St000616 The inversion index of a permutation.

St000619 The number of cyclic descents of a permutation.

St000622 The number of occurrences of the patterns 2143 or 4231 in a permutation.

St000623 The number of occurrences of the pattern 52341 in a permutation.

St000624 The normalized sum of the minimal distances to a greater element.

St000625 The sum of the minimal distances to a greater element.

St000638 The number of up-down runs of a permutation.

St000646 The number of big ascents of a permutation.

St000647 The number of big descents of a permutation.

St000648 The number of 2-excedences of a permutation.

St000649 The number of 3-excedences of a permutation.

St000650 The number of 3-rises of a permutation.

St000651 The maximal size of a rise in a permutation.

St000652 The maximal difference between successive positions of a permutation.

St000653 The last descent of a permutation.

St000654 The first descent of a permutation.

St000662 The staircase size of the code of a permutation.

St000663 The number of right floats of a permutation.

St000664 The number of right ropes of a permutation.

St000665 The number of rafts of a permutation.

St000666 The number of right tethers of a permutation.

St000669 The number of permutations obtained by switching ascents or descents of size 2.

St000670 The reversal length of a permutation.

St000672 The number of minimal elements in Bruhat order not less than the permutation.

St000673 The size of the support of a permutation.

St000677 The standardized bi-alternating inversion number of a permutation.

St000690 The size of the conjugacy class of a permutation.

St000692 Babson and Steingrímsson's statistic stat of a permutation.

St000694 The number of affine bounded permutations that project to a given permutation.

St000696 The number of cycles in the breakpoint graph of a permutation.

St000702 The number of weak deficiencies of a permutation.

St000703 The number of deficiencies of a permutation.

St000709 The number of occurrences of 14-2-3 or 14-3-2.

St000710 The number of big deficiencies of a permutation.

St000711 The number of big exceedences of a permutation.

St000724 The label of the leaf of the path following the smaller label in the increasing b....

St000725 The smallest label of a leaf of the increasing binary tree associated to a permut....

St000726 The normalized sum of the leaf labels of the increasing binary tree associated to....

St000727 The largest label of a leaf in the binary search tree associated with the permuta....

St000731 The number of double exceedences of a permutation.

St000732 The number of double deficiencies of a permutation.

St000740 The last entry of a permutation.

St000742 The number of big ascents of a permutation after adding the value $\pi(0) = 0$.

St000750 The number of occurrences of the pattern 4213 in a permutation.

St000751 The number of occurrences of either of the pattern 2143 or 2143 in a permutation.....

St000756 The sum of the positions of the left to right maxima of a permutation.

St000779 The tier of a permutation.

St000794 The mak of a permutation.

St000795 The mad of a permutation.

St000796 The stat' of a permutation.

St000797 The stat`` of a permutation.

St000798 The makl of a permutation.

St000799 The number of occurrences of the vincular pattern |213 in a permutation.

St000800 The number of occurrences of the vincular pattern |231 in a permutation.

St000801 The number of occurrences of the vincular pattern |312 in a permutation.

St000802 The number of occurrences of the vincular pattern |321 in a permutation.

St000803 The number of occurrences of the vincular pattern |132 in a permutation.

St000804 The number of occurrences of the vincular pattern |123 in a permutation.

St000809 The reduced reflection length of the permutation.

St000824 The sum of the number of descents and the number of recoils of a permutation.

St000825 The sum of the number of major and the inverse major index of a permutation.

St000828 The spearman's rho of a permutation and the identity permutation.

St000829 The Ulam distance of a permutation to the identity permutation.

St000830 The total displacement of a permutation.

St000831 The number of indices that are either descents or recoils.

St000832 The number of permutations obtained by reversing blocks of three consecutive numb....

St000833 The comajor index of a permutation.

St000834 The number of right outer peaks of a permutation.

St000836 The number of descents of distance 2 of a permutation.

St000837 The number of ascents of distance 2 of a permutation.

St000842 The breadth of a permutation.

St000844 The size of the largest block in the direct sum decomposition of a permutation.

St000862 The number of parts of the shifted shape of a permutation.

St000863 The length of the first row of the shifted shape of a permutation.

St000864 The number of circled entries of the shifted recording tableau of a permutation.

St000866 The number of admissible inversions of a permutation in the sense of Shareshian-W....

St000868 The aid statistic in the sense of Shareshian-Wachs.

St000871 The number of very big ascents of a permutation.

St000872 The number of very big descents of a permutation.

St000873 The aix statistic of a permutation.

St000879 The number of long braid edges in the graph of braid moves of a permutation.

St000880 The number of connected components of long braid edges in the graph of braid move....

St000881 The number of short braid edges in the graph of braid moves of a permutation.

St000882 The number of connected components of short braid edges in the graph of braid mov....

St000883 The number of longest increasing subsequences of a permutation.

St000884 The number of isolated descents of a permutation.

St000886 The number of permutations with the same antidiagonal sums.

St000887 The maximal number of nonzero entries on a diagonal of a permutation matrix.

St000891 The number of distinct diagonal sums of a permutation matrix.

St000923 The minimal number with no two order isomorphic substrings of this length in a pe....

St000002 The number of occurrences of the pattern 123 in a permutation.

St000004 The major index of a permutation.

St000007 The number of saliances of the permutation.

St000018 The number of inversions of a permutation.

St000019 The cardinality of the complement of the connectivity set.

St000020 The rank of the permutation.

St000021 The number of descents of a permutation.

St000022 The number of fixed points of a permutation.

St000023 The number of inner peaks of a permutation.

St000028 The number of stack-sorts needed to sort a permutation.

St000029 The depth of a permutation.

St000030 The sum of the descent differences of a permutations.

St000031 The number of cycles in the cycle decomposition of a permutation.

St000033 The number of permutations greater than or equal to the given permutation in (str....

St000034 The maximum defect over any reduced expression for a permutation and any subexpre....

St000035 The number of left outer peaks of a permutation.

St000036 The evaluation of the Kazhdan-Lusztig polynomial $P(id,w)$ for each permutation $w$ i....

St000037 The sign of a permutation.

St000039 The number of crossings of a permutation.

St000040 The number of regions of inversion arrangement of a permutation.

St000054 The first entry of the permutation.

St000055 The inversion sum of a permutation.

St000056 The decomposition (or block) number of a permutation.

St000058 The order of a permutation.

St000060 The greater neighbor of the maximum.

St000062 The length of the longest increasing subsequence of the permutation.

St000064 The number of one-box pattern of a permutation.

St000078 The number of alternating sign matrices whose left key is the permutation.

St000092 The number of outer peaks of a permutation.

St000099 The number of valleys of a permutation, including the boundary.

St000109 The number of elements less than or equal to the given element in Bruhat order.

St000110 The number of permutations less than or equal to given permutation in left weak o....

St000111 The sum of the descent tops (or Genocchi descents) of a permutation.

St000119 The number of occurrences of the pattern 321 in a permutation.

St000123 The difference in Coxeter length of a permutation and its image under the Simion-....

St000124 The cardinality of the preimage of the Simion-Schmidt map.

St000133 The "bounce" of a permutation.

St000141 The maximum drop size of a permutation.

St000153 The number of adjacent cycles of a permutation.

St000154 The sum of the descent bottoms of a permutation.

St000155 The number of exceedances (also excedences) of a permutation.

St000156 The Denert index of a permutation.

St000162 The number of nontrivial cycles of a permutation $\pi$ in its cycle decomposition.

St000209 Maximum difference of elements in cycles.

St000210 Minimum over maximum difference of elements in cycles.

St000213 The number of weak exceedances (also weak excedences) of a permutation.

St000214 The number of adjacencies (or small descents) of a permutation.

St000215 The number of adjacencies of a permutation, 0 appended.

St000216 The absolute length of a permutation.

St000217 The number of occurrences of the pattern 312 in a permutation.

St000218 The number of occurrences of the pattern 213 in a permutation.

St000219 The number of occurrences of the pattern 231 in a permutation.

St000220 The number of occurrences of the pattern 132 in a permutation.

St000221 The number of strong fixed points of a permutation.

St000222 The number of alignments of a permutation

St000223 The number of nestings of a permutation

St000224 The sorting index of a permutation.

St000226 The convexity of a permutation.

St000234 The number of global ascents of a permutation.

St000235 The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically.

St000236 The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

St000237 The number of indices $i$ such that $\pi_i=i+1$.

St000238 The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.

St000239 The number of indices $i$ such that $\pi_i \in \{i,i+1\}$.

St000240 The number of indices $i$ for which $\pi_i \neq i+1$.

St000241 The number of indices $i$ such that $\pi_i = i+1$ considered cyclically.

St000242 The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically.....

St000243 The number of cyclic valleys and cyclic peaks of a permutation.

St000245 The number of ascents of a permutation.

St000246 The number of non-inversions of a permutation.

St000255 The number of reduced Kogan faces with the permutation as type.

St000279 The size of the preimage of the map 'cycle-as-one-line notation' from Permutation....

St000280 The size of the preimage of the map 'to labelling permutation' from Parking funct....

St000304 The load of a permutation.

St000305 The inverse major index of a permutation.

St000308 The height of the tree associated to a permutation.

St000314 The number of left-to-right-maxima of a permutation.

St000316 The number of non-left-to-right-maxima of a permutation.

St000317 The cycle descent number of a permutation.

St000324 The shape of the tree associated to a permutation.

St000325 The width of the tree associated to a permutation.

St000333 The dez statistic, the number of descents of a permutation after replacing fixed ....

St000334 The maz index, the major index of a permutation after replacing fixed points by z....

St000337 The lec statistic, the sum of the inversion numbers of the hook factors of a perm....

St000338 The number of pixed points of a permutation.

St000339 The maf index of a permutation.

St000341 The non-inversion sum of a permutation.

St000342 The cosine of a permutation.

St000352 The Elizalde-Pak rank of a permutation.

St000353 The number of inner valleys of a permutation.

St000354 The number of recoils of a permutation.

St000355 The number of occurrences of the pattern 21-3.

St000356 The number of occurrences of the pattern 13-2.

St000357 The number of occurrences of the pattern 12-3.

St000358 The number of occurrences of the pattern 31-2.

St000359 The number of occurrences of the pattern 23-1.

St000360 The number of occurrences of the pattern 32-1.

St000365 The number of double ascents of a permutation.

St000366 The number of double descents of a permutation.

St000367 The number of simsun double descents of a permutation.

St000371 The number of positions mid points of decreasing subsequences of length 3 in a pe....

St000372 The number of positions of mid points of increasing subsequences of length 3 in a....

St000373 The number of weak exceedences of a permutation that are also mid-points of a dec....

St000374 The number of non weak exceedences of a permutation that are also not mid-points ....

St000375 The number of non weak exceedences of a permutation that are mid-points of a decr....

St000401 The size of the symmetry class of a permutation.

St000402 Half the size of the symmetry class of a permutation.

St000404 The number of occurrences of the pattern 3241 or of the pattern 4231 in a permuta....

St000405 The number of occurrences of the pattern 1324 in a permutation.

St000406 The number of occurrences of the pattern 3241 in a permutation.

St000407 The number of occurrences of the pattern 2143 in a permutation.

St000408 The number of occurrences of the pattern 4231 in a permutation.

St000423 The number of occurrences of the pattern 123 or of the pattern 132 in a permutati....

St000424 The number of occurrences of the pattern 132 or of the pattern 231 in a permutati....

St000425 The number of occurrences of the pattern 132 or of the pattern 213 in a permutati....

St000426 The number of occurrences of the pattern 132 or of the pattern 312 in a permutati....

St000427 The number of occurrences of the pattern 123 or of the pattern 231 in a permutati....

St000428 The number of occurrences of the pattern 123 or of the pattern 213 in a permutati....

St000429 The number of occurrences of the pattern 123 or of the pattern 321 in a permutati....

St000430 The number of occurrences of the pattern 123 or of the pattern 312 in a permutati....

St000431 The number of occurrences of the pattern 213 or of the pattern 321 in a permutati....

St000432 The number of occurrences of the pattern 231 or of the pattern 312 in a permutati....

St000433 The number of occurrences of the pattern 132 or of the pattern 321 in a permutati....

St000434 The number of occurrences of the pattern 213 or of the pattern 312 in a permutati....

St000435 The number of occurrences of the pattern 213 or of the pattern 231 in a permutati....

St000436 The number of occurrences of the pattern 231 or of the pattern 321 in a permutati....

St000437 The number of occurrences of the pattern 312 or of the pattern 321 in a permutati....

St000440 The number of occurrences of the pattern 4132 or of the pattern 4231 in a permuta....

St000441 The number of successions (or small ascents) of a permutation.

St000446 The disorder of a permutation.

St000451 The length of the longest pattern of the form k 1 2.

St000457 The number of occurrences of one of the patterns 132, 213 or 321 in a permutation....

St000458 The number of permutations obtained by switching adjacencies or successions.

St000461 The rix statistic of a permutation.

St000462 The major index minus the number of excedences of a permutation.

St000463 The number of admissible inversions of a permutation.

St000470 The number of runs in a permutation.

St000471 The sum of the ascent tops of a permutation.

St000472 The sum of the ascent bottoms of a permutation.

St000483 The number of times a permutation switches from increasing to decreasing or decre....

St000484 The sum of St000483 over all subsequences of length at least three.

St000485 The length of the longest cycle of a permutation.

St000486 The number of cycles of length at least 3 of a permutation.

St000487 The length of the shortest cycle of a permutation.

St000488 The number of cycle of a permutation of length at most 2.

St000489 The number of cycle of a permutation of length at most 3.

St000494 The number of inversions of distance at most 3 of a permutation.

St000495 The number of inversions of distance at most 2 of a permutation.

St000500 Eigenvalues of the random-to-random operator acting on the regular representation....

St000501 The size of the first part in the decomposition of a permutation.

St000516 The number of stretching pairs of a permutation.

St000520 The number of subsequences of a permutation which are not order-isomorphic.

St000530 The number of permutations with the same descent word as the given permutation.

St000534 The number of 2-rises of a permutation.

St000538 The number of even inversions of a permutation.

St000539 The number of odd inversions of a permutation.

St000541 The number of indices greater than or equal to 2 of a permutation such that all s....

St000542 The number of left-to-right-minima of a permutation.

St000545 The number of parabolic double cosets with minimal element being the given permut....

St000546 The number of global descents of a permutation.

St000570 The Edelman-Greene number of a permutation.

St000616 The inversion index of a permutation.

St000619 The number of cyclic descents of a permutation.

St000622 The number of occurrences of the patterns 2143 or 4231 in a permutation.

St000623 The number of occurrences of the pattern 52341 in a permutation.

St000624 The normalized sum of the minimal distances to a greater element.

St000625 The sum of the minimal distances to a greater element.

St000638 The number of up-down runs of a permutation.

St000646 The number of big ascents of a permutation.

St000647 The number of big descents of a permutation.

St000648 The number of 2-excedences of a permutation.

St000649 The number of 3-excedences of a permutation.

St000650 The number of 3-rises of a permutation.

St000651 The maximal size of a rise in a permutation.

St000652 The maximal difference between successive positions of a permutation.

St000653 The last descent of a permutation.

St000654 The first descent of a permutation.

St000662 The staircase size of the code of a permutation.

St000663 The number of right floats of a permutation.

St000664 The number of right ropes of a permutation.

St000665 The number of rafts of a permutation.

St000666 The number of right tethers of a permutation.

St000669 The number of permutations obtained by switching ascents or descents of size 2.

St000670 The reversal length of a permutation.

St000672 The number of minimal elements in Bruhat order not less than the permutation.

St000673 The size of the support of a permutation.

St000677 The standardized bi-alternating inversion number of a permutation.

St000690 The size of the conjugacy class of a permutation.

St000692 Babson and Steingrímsson's statistic stat of a permutation.

St000694 The number of affine bounded permutations that project to a given permutation.

St000696 The number of cycles in the breakpoint graph of a permutation.

St000702 The number of weak deficiencies of a permutation.

St000703 The number of deficiencies of a permutation.

St000709 The number of occurrences of 14-2-3 or 14-3-2.

St000710 The number of big deficiencies of a permutation.

St000711 The number of big exceedences of a permutation.

St000724 The label of the leaf of the path following the smaller label in the increasing b....

St000725 The smallest label of a leaf of the increasing binary tree associated to a permut....

St000726 The normalized sum of the leaf labels of the increasing binary tree associated to....

St000727 The largest label of a leaf in the binary search tree associated with the permuta....

St000731 The number of double exceedences of a permutation.

St000732 The number of double deficiencies of a permutation.

St000740 The last entry of a permutation.

St000742 The number of big ascents of a permutation after adding the value $\pi(0) = 0$.

St000750 The number of occurrences of the pattern 4213 in a permutation.

St000751 The number of occurrences of either of the pattern 2143 or 2143 in a permutation.....

St000756 The sum of the positions of the left to right maxima of a permutation.

St000779 The tier of a permutation.

St000794 The mak of a permutation.

St000795 The mad of a permutation.

St000796 The stat' of a permutation.

St000797 The stat`` of a permutation.

St000798 The makl of a permutation.

St000799 The number of occurrences of the vincular pattern |213 in a permutation.

St000800 The number of occurrences of the vincular pattern |231 in a permutation.

St000801 The number of occurrences of the vincular pattern |312 in a permutation.

St000802 The number of occurrences of the vincular pattern |321 in a permutation.

St000803 The number of occurrences of the vincular pattern |132 in a permutation.

St000804 The number of occurrences of the vincular pattern |123 in a permutation.

St000809 The reduced reflection length of the permutation.

St000824 The sum of the number of descents and the number of recoils of a permutation.

St000825 The sum of the number of major and the inverse major index of a permutation.

St000828 The spearman's rho of a permutation and the identity permutation.

St000829 The Ulam distance of a permutation to the identity permutation.

St000830 The total displacement of a permutation.

St000831 The number of indices that are either descents or recoils.

St000832 The number of permutations obtained by reversing blocks of three consecutive numb....

St000833 The comajor index of a permutation.

St000834 The number of right outer peaks of a permutation.

St000836 The number of descents of distance 2 of a permutation.

St000837 The number of ascents of distance 2 of a permutation.

St000842 The breadth of a permutation.

St000844 The size of the largest block in the direct sum decomposition of a permutation.

St000862 The number of parts of the shifted shape of a permutation.

St000863 The length of the first row of the shifted shape of a permutation.

St000864 The number of circled entries of the shifted recording tableau of a permutation.

St000866 The number of admissible inversions of a permutation in the sense of Shareshian-W....

St000868 The aid statistic in the sense of Shareshian-Wachs.

St000871 The number of very big ascents of a permutation.

St000872 The number of very big descents of a permutation.

St000873 The aix statistic of a permutation.

St000879 The number of long braid edges in the graph of braid moves of a permutation.

St000880 The number of connected components of long braid edges in the graph of braid move....

St000881 The number of short braid edges in the graph of braid moves of a permutation.

St000882 The number of connected components of short braid edges in the graph of braid mov....

St000883 The number of longest increasing subsequences of a permutation.

St000884 The number of isolated descents of a permutation.

St000886 The number of permutations with the same antidiagonal sums.

St000887 The maximal number of nonzero entries on a diagonal of a permutation matrix.

St000891 The number of distinct diagonal sums of a permutation matrix.

St000923 The minimal number with no two order isomorphic substrings of this length in a pe....

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

St000068
The number of minimal elements in a poset.

St000069 The number of maximal elements of a poset.

St000070 The number of antichains in a poset.

St000071 The number of maximal chains in a poset.

St000080 The rank of the poset.

St000100 The number of linear extensions of a poset.

St000104 The number of facets in the order polytope of this poset.

St000151 The number of facets in the chain polytope of the poset.

St000180 The number of chains of a poset.

St000181 The number of connected components of the Hasse diagram for the poset.

St000189 The number of elements in the poset.

St000281 The size of the preimage of the map 'to poset' from Binary trees to Posets.

St000282 The size of the preimage of the map 'to poset' from Ordered trees to Posets.

St000298 The order dimension or Dushnik-Miller dimension of a poset.

St000307 The number of rowmotion orbits of a poset.

St000327 The number of cover relations in a poset.

St000524 The number of posets with the same order polynomial.

St000525 The number of posets with the same zeta polynomial.

St000526 The number of posets with combinatorially isomorphic order polytopes.

St000527 The width of the poset.

St000528 The height of a poset.

St000550 The number of modular elements of a lattice.

St000551 The number of left modular elements of a lattice.

St000632 The jump number of the poset.

St000633 The size of the automorphism group of a poset.

St000634 The number of endomorphisms of a poset.

St000635 The number of strictly order preserving maps of a poset into itself.

St000639 The number of relations in a poset.

St000640 The rank of the largest boolean interval in a poset.

St000641 The number of non-empty boolean intervals in a poset.

St000642 The size of the smallest orbit of antichains under Panyushev complementation.

St000643 The size of the largest orbit of antichains under Panyushev complementation.

St000656 The number of cuts of a poset.

St000680 The Grundy value for Hackendot on posets.

St000717 The number of ordinal summands of a poset.

St000845 The maximal number of elements covered by an element in a poset.

St000846 The maximal number of elements covering an element of a poset.

St000848 The balance constant multiplied with the number of linear extensions of a poset.

St000849 The number of 1/3-balanced pairs in a poset.

St000850 The number of 1/2-balanced pairs in a poset.

St000906 The length of the shortest maximal chain in a poset.

St000907 The number of maximal antichains of minimal length in a poset.

St000908 The length of the shortest maximal antichain in a poset.

St000909 The number of maximal chains of maximal size in a poset.

St000910 The number of maximal chains of minimal length in a poset.

St000911 The number of maximal antichains of maximal size in a poset.

St000912 The number of maximal antichains in a poset.

St000914 The sum of the values of the Möbius function of a poset.

St000069 The number of maximal elements of a poset.

St000070 The number of antichains in a poset.

St000071 The number of maximal chains in a poset.

St000080 The rank of the poset.

St000100 The number of linear extensions of a poset.

St000104 The number of facets in the order polytope of this poset.

St000151 The number of facets in the chain polytope of the poset.

St000180 The number of chains of a poset.

St000181 The number of connected components of the Hasse diagram for the poset.

St000189 The number of elements in the poset.

St000281 The size of the preimage of the map 'to poset' from Binary trees to Posets.

St000282 The size of the preimage of the map 'to poset' from Ordered trees to Posets.

St000298 The order dimension or Dushnik-Miller dimension of a poset.

St000307 The number of rowmotion orbits of a poset.

St000327 The number of cover relations in a poset.

St000524 The number of posets with the same order polynomial.

St000525 The number of posets with the same zeta polynomial.

St000526 The number of posets with combinatorially isomorphic order polytopes.

St000527 The width of the poset.

St000528 The height of a poset.

St000550 The number of modular elements of a lattice.

St000551 The number of left modular elements of a lattice.

St000632 The jump number of the poset.

St000633 The size of the automorphism group of a poset.

St000634 The number of endomorphisms of a poset.

St000635 The number of strictly order preserving maps of a poset into itself.

St000639 The number of relations in a poset.

St000640 The rank of the largest boolean interval in a poset.

St000641 The number of non-empty boolean intervals in a poset.

St000642 The size of the smallest orbit of antichains under Panyushev complementation.

St000643 The size of the largest orbit of antichains under Panyushev complementation.

St000656 The number of cuts of a poset.

St000680 The Grundy value for Hackendot on posets.

St000717 The number of ordinal summands of a poset.

St000845 The maximal number of elements covered by an element in a poset.

St000846 The maximal number of elements covering an element of a poset.

St000848 The balance constant multiplied with the number of linear extensions of a poset.

St000849 The number of 1/3-balanced pairs in a poset.

St000850 The number of 1/2-balanced pairs in a poset.

St000906 The length of the shortest maximal chain in a poset.

St000907 The number of maximal antichains of minimal length in a poset.

St000908 The length of the shortest maximal antichain in a poset.

St000909 The number of maximal chains of maximal size in a poset.

St000910 The number of maximal chains of minimal length in a poset.

St000911 The number of maximal antichains of maximal size in a poset.

St000912 The number of maximal antichains in a poset.

St000914 The sum of the values of the Möbius function of a poset.

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

St000101
The cocharge of a semistandard tableau.

St000102 The charge of a semistandard tableau.

St000103 The sum of the entries of a semistandard tableau.

St000112 The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....

St000116 The major index of a semistandard tableau obtained by standardizing.

St000170 The trace of a semistandard tableau.

St000173 The segment statistic of a semistandard tableau.

St000174 The flush statistic of a semistandard tableau.

St000736 The last entry in the first row of a semistandard tableau.

St000737 The last entry on the main diagonal of a semistandard tableau.

St000739 The first entry in the last row of a semistandard tableau.

St000102 The charge of a semistandard tableau.

St000103 The sum of the entries of a semistandard tableau.

St000112 The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....

St000116 The major index of a semistandard tableau obtained by standardizing.

St000170 The trace of a semistandard tableau.

St000173 The segment statistic of a semistandard tableau.

St000174 The flush statistic of a semistandard tableau.

St000736 The last entry in the first row of a semistandard tableau.

St000737 The last entry on the main diagonal of a semistandard tableau.

St000739 The first entry in the last row of a semistandard tableau.

**Set partitions**(93 statistics) # partition like objects

St000105
The number of blocks in the set partition.

St000163 The size of the orbit of the set partition under rotation.

St000211 The rank of the set partition.

St000229 Sum of the difference between the maximal and the minimal elements of the blocks ....

St000230 Sum of the minimal elements of the blocks of a set partition.

St000231 Sum of the maximal elements of the blocks of a set partition.

St000232 The number of crossings of a set partition.

St000233 The number of nestings of a set partition.

St000247 The number of singleton blocks of a set partition.

St000248 The number of anti-singletons of a set partition.

St000249 The number of singletons (St000247) plus the number of antisingletons (St000248) ....

St000250 The number of blocks (St000105) plus the number of antisingletons (St000248) of a....

St000251 The number of nonsingleton blocks of a set partition.

St000253 The crossing number of a set partition.

St000254 The nesting number of a set partition.

St000490 The intertwining number of a set partition.

St000491 The number of inversions of a set partition.

St000492 The rob statistic of a set partition.

St000493 The los statistic of a set partition.

St000496 The rcs statistic of a set partition.

St000497 The lcb statistic of a set partition.

St000498 The lcs statistic of a set partition.

St000499 The rcb statistic of a set partition.

St000502 The number of successions of a set partitions.

St000503 The maximal difference between two elements in a common block.

St000504 The cardinality of the first block of a set partition.

St000505 The biggest entry in the block containing the 1.

St000554 The number of occurrences of the pattern {{1,2},{3}} in a set partition.

St000555 The number of occurrences of the pattern {{1,3},{2}} in a set partition.

St000556 The number of occurrences of the pattern {{1},{2,3}} in a set partition.

St000557 The number of occurrences of the pattern {{1},{2},{3}} in a set partition.

St000558 The number of occurrences of the pattern {{1,2}} in a set partition.

St000559 The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.

St000560 The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.

St000561 The number of occurrences of the pattern {{1,2,3}} in a set partition.

St000562 The number of internal points of a set partition.

St000563 The number of overlapping pairs of blocks of a set partition.

St000564 The number of occurrences of the pattern {{1},{2}} in a set partition.

St000565 The major index of a set partition.

St000572 The dimension exponent of a set partition.

St000573 The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 ....

St000574 The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a ....

St000575 The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element ....

St000576 The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a ....

St000577 The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.....

St000578 The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.

St000579 The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.....

St000580 The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is m....

St000581 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is ma....

St000582 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is ma....

St000583 The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1,2 ar....

St000584 The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is m....

St000585 The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) a....

St000586 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.

St000587 The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.

St000588 The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 i....

St000589 The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) a....

St000590 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is ma....

St000591 The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.

St000592 The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.

St000593 The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.

St000594 The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3....

St000595 The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.

St000596 The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is m....

St000597 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) a....

St000598 The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is....

St000599 The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive....

St000600 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) a....

St000601 The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3....

St000602 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.

St000603 The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.

St000604 The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is m....

St000605 The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) a....

St000606 The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3....

St000607 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is ma....

St000608 The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 i....

St000609 The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

St000610 The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.

St000611 The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.

St000612 The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) a....

St000613 The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is ma....

St000614 The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is ma....

St000615 The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.

St000695 The number of blocks in the first part of the atomic decomposition of a set parti....

St000728 The dimension of a set partition.

St000729 The minimal arc length of a set partition.

St000730 The maximal arc length of a set partition.

St000747 A variant of the major index of a set partition.

St000748 The major index of the permutation obtained by flattening the set partition.

St000793 The length of the longest partition in the vacillating tableau corresponding to a....

St000823 The number of unsplittable factors of the set partition.

St000839 The largest opener of a set partition.

St000925 The number of topologically connected components of a set partition.

St000163 The size of the orbit of the set partition under rotation.

St000211 The rank of the set partition.

St000229 Sum of the difference between the maximal and the minimal elements of the blocks ....

St000230 Sum of the minimal elements of the blocks of a set partition.

St000231 Sum of the maximal elements of the blocks of a set partition.

St000232 The number of crossings of a set partition.

St000233 The number of nestings of a set partition.

St000247 The number of singleton blocks of a set partition.

St000248 The number of anti-singletons of a set partition.

St000249 The number of singletons (St000247) plus the number of antisingletons (St000248) ....

St000250 The number of blocks (St000105) plus the number of antisingletons (St000248) of a....

St000251 The number of nonsingleton blocks of a set partition.

St000253 The crossing number of a set partition.

St000254 The nesting number of a set partition.

St000490 The intertwining number of a set partition.

St000491 The number of inversions of a set partition.

St000492 The rob statistic of a set partition.

St000493 The los statistic of a set partition.

St000496 The rcs statistic of a set partition.

St000497 The lcb statistic of a set partition.

St000498 The lcs statistic of a set partition.

St000499 The rcb statistic of a set partition.

St000502 The number of successions of a set partitions.

St000503 The maximal difference between two elements in a common block.

St000504 The cardinality of the first block of a set partition.

St000505 The biggest entry in the block containing the 1.

St000554 The number of occurrences of the pattern {{1,2},{3}} in a set partition.

St000555 The number of occurrences of the pattern {{1,3},{2}} in a set partition.

St000556 The number of occurrences of the pattern {{1},{2,3}} in a set partition.

St000557 The number of occurrences of the pattern {{1},{2},{3}} in a set partition.

St000558 The number of occurrences of the pattern {{1,2}} in a set partition.

St000559 The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.

St000560 The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.

St000561 The number of occurrences of the pattern {{1,2,3}} in a set partition.

St000562 The number of internal points of a set partition.

St000563 The number of overlapping pairs of blocks of a set partition.

St000564 The number of occurrences of the pattern {{1},{2}} in a set partition.

St000565 The major index of a set partition.

St000572 The dimension exponent of a set partition.

St000573 The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 ....

St000574 The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a ....

St000575 The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element ....

St000576 The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a ....

St000577 The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.....

St000578 The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.

St000579 The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.....

St000580 The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is m....

St000581 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is ma....

St000582 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is ma....

St000583 The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1,2 ar....

St000584 The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is m....

St000585 The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) a....

St000586 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.

St000587 The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.

St000588 The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 i....

St000589 The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) a....

St000590 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is ma....

St000591 The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.

St000592 The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.

St000593 The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.

St000594 The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3....

St000595 The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.

St000596 The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is m....

St000597 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) a....

St000598 The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is....

St000599 The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive....

St000600 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) a....

St000601 The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3....

St000602 The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.

St000603 The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.

St000604 The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is m....

St000605 The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) a....

St000606 The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3....

St000607 The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is ma....

St000608 The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 i....

St000609 The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

St000610 The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.

St000611 The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.

St000612 The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) a....

St000613 The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is ma....

St000614 The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is ma....

St000615 The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.

St000695 The number of blocks in the first part of the atomic decomposition of a set parti....

St000728 The dimension of a set partition.

St000729 The minimal arc length of a set partition.

St000730 The maximal arc length of a set partition.

St000747 A variant of the major index of a set partition.

St000748 The major index of the permutation obtained by flattening the set partition.

St000793 The length of the longest partition in the vacillating tableau corresponding to a....

St000823 The number of unsplittable factors of the set partition.

St000839 The largest opener of a set partition.

St000925 The number of topologically connected components of a set partition.

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

St000009
The charge of a standard tableau.

St000016 The number of attacking pairs of a standard tableau.

St000017 The number of inversions of a standard tableau.

St000057 The Shynar inversion number of a standard tableau.

St000059 The inversion number of a standard tableau as defined by Haglund and Stevens.

St000075 The orbit size of a standard tableau under promotion.

St000157 The number of descents of a standard tableau.

St000169 The cocharge of a standard tableau.

St000330 The (standard) major index of a standard tableau.

St000336 The leg major index of a standard tableau.

St000507 The number of ascents of a standard tableau.

St000508 Eigenvalues of the random-to-random operator acting on a simple module.

St000693 The modular (standard) major index of a standard tableau.

St000733 The row containing the largest entry of a standard tableau.

St000734 The last entry in the first row of a standard tableau.

St000735 The last entry on the main diagonal of a standard tableau.

St000738 The first entry in the last row of a standard tableau.

St000743 The number of entries in a standard Young tableau such that the next integer is a....

St000744 The length of the path to the largest entry in a standard Young tableau.

St000745 The index of the last row whose first entry is the row number in a standard Young....

St000016 The number of attacking pairs of a standard tableau.

St000017 The number of inversions of a standard tableau.

St000057 The Shynar inversion number of a standard tableau.

St000059 The inversion number of a standard tableau as defined by Haglund and Stevens.

St000075 The orbit size of a standard tableau under promotion.

St000157 The number of descents of a standard tableau.

St000169 The cocharge of a standard tableau.

St000330 The (standard) major index of a standard tableau.

St000336 The leg major index of a standard tableau.

St000507 The number of ascents of a standard tableau.

St000508 Eigenvalues of the random-to-random operator acting on a simple module.

St000693 The modular (standard) major index of a standard tableau.

St000733 The row containing the largest entry of a standard tableau.

St000734 The last entry in the first row of a standard tableau.

St000735 The last entry on the main diagonal of a standard tableau.

St000738 The first entry in the last row of a standard tableau.

St000743 The number of entries in a standard Young tableau such that the next integer is a....

St000744 The length of the path to the largest entry in a standard Young tableau.

St000745 The index of the last row whose first entry is the row number in a standard Young....