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Your data matches 409 different statistics following compositions of up to 3 maps.
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Matching statistic: St000003
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St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 1 = 0 + 1
[1,1]
=> 1 = 0 + 1
[3]
=> 1 = 0 + 1
[2,1]
=> 2 = 1 + 1
[1,1,1]
=> 1 = 0 + 1
[4]
=> 1 = 0 + 1
[3,1]
=> 3 = 2 + 1
[2,2]
=> 2 = 1 + 1
[2,1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> 1 = 0 + 1
[5]
=> 1 = 0 + 1
[1,1,1,1,1]
=> 1 = 0 + 1
[6]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St001780
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St001780: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 1 = 0 + 1
[1,1]
=> 1 = 0 + 1
[3]
=> 1 = 0 + 1
[2,1]
=> 2 = 1 + 1
[1,1,1]
=> 1 = 0 + 1
[4]
=> 1 = 0 + 1
[3,1]
=> 3 = 2 + 1
[2,2]
=> 2 = 1 + 1
[2,1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> 1 = 0 + 1
[5]
=> 1 = 0 + 1
[1,1,1,1,1]
=> 1 = 0 + 1
[6]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St000057
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Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> 0
[1,1]
=> [[1],[2]]
=> 0
[3]
=> [[1,2,3]]
=> 0
[2,1]
=> [[1,2],[3]]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> 0
[4]
=> [[1,2,3,4]]
=> 0
[3,1]
=> [[1,2,3],[4]]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> 1
[2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[5]
=> [[1,2,3,4,5]]
=> 0
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[6]
=> [[1,2,3,4,5,6]]
=> 0
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
Description
The Shynar inversion number of a standard tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000075
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(load all 42 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000075: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> 1 = 0 + 1
[1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[3]
=> [[1,2,3]]
=> 1 = 0 + 1
[2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[6]
=> [[1,2,3,4,5,6]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1 = 0 + 1
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St001595
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Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001595: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[2],[]]
=> 1 = 0 + 1
[1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3]
=> [[3],[]]
=> 1 = 0 + 1
[2,1]
=> [[2,1],[]]
=> 2 = 1 + 1
[1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[4]
=> [[4],[]]
=> 1 = 0 + 1
[3,1]
=> [[3,1],[]]
=> 3 = 2 + 1
[2,2]
=> [[2,2],[]]
=> 2 = 1 + 1
[2,1,1]
=> [[2,1,1],[]]
=> 3 = 2 + 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[5]
=> [[5],[]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[6]
=> [[6],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 1 = 0 + 1
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St001929
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(load all 20 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001929: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001929: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
Description
The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path.
Matching statistic: St000293
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Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> 0 => 0
[1,1]
=> [[1],[2]]
=> 1 => 0
[3]
=> [[1,2,3]]
=> 00 => 0
[2,1]
=> [[1,3],[2]]
=> 10 => 1
[1,1,1]
=> [[1],[2],[3]]
=> 11 => 0
[4]
=> [[1,2,3,4]]
=> 000 => 0
[3,1]
=> [[1,3,4],[2]]
=> 100 => 2
[2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 0
[5]
=> [[1,2,3,4,5]]
=> 0000 => 0
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 0
[6]
=> [[1,2,3,4,5,6]]
=> 00000 => 0
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 11111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St000426
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Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000426: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000426: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 312 in a permutation.
Matching statistic: St000427
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Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000427: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000427: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => 0
Description
The number of occurrences of the pattern 123 or of the pattern 231 in a permutation.
Matching statistic: St000433
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Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000433: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000433: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.
The following 399 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000682The Grundy value of Welter's game on a binary word. St000747A variant of the major index of a set partition. St000849The number of 1/3-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000001The number of reduced words for a permutation. St000100The number of linear extensions of a poset. St000626The minimal period of a binary word. St001268The size of the largest ordinal summand in the poset. St001313The number of Dyck paths above the lattice path given by a binary word. St001779The order of promotion on the set of linear extensions of a poset. St000081The number of edges of a graph. St000171The degree of the graph. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000271The chromatic index of a graph. St000290The major index of a binary word. St000295The length of the border of a binary word. St000355The number of occurrences of the pattern 21-3. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000691The number of changes of a binary word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001175The size of a partition minus the hook length of the base cell. St001306The number of induced paths on four vertices in a graph. St001341The number of edges in the center of a graph. St001377The major index minus the number of inversions of a permutation. St001479The number of bridges of a graph. St001485The modular major index of a binary word. St001512The minimum rank of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001584The area statistic between a Dyck path and its bounce path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001869The maximum cut size of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001910The height of the middle non-run of a Dyck path. St001911A descent variant minus the number of inversions. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000078The number of alternating sign matrices whose left key is the permutation. St000086The number of subgraphs. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000299The number of nonisomorphic vertex-induced subtrees. St000388The number of orbits of vertices of a graph under automorphisms. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000446The disorder of a permutation. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000530The number of permutations with the same descent word as the given permutation. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000883The number of longest increasing subsequences of a permutation. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001415The length of the longest palindromic prefix of a binary word. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001884The number of borders of a binary word. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000321The number of integer partitions of n that are dominated by an integer partition. St000378The diagonal inversion number of an integer partition. St000439The position of the first down step of a Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St001486The number of corners of the ribbon associated with an integer composition. St000452The number of distinct eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000119The number of occurrences of the pattern 321 in a permutation. St000209Maximum difference of elements in cycles. St000539The number of odd inversions of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. St000956The maximal displacement of a permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000454The largest eigenvalue of a graph if it is integral. St000658The number of rises of length 2 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001411The number of patterns 321 or 3412 in a permutation. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St001657The number of twos in an integer partition. St001811The Castelnuovo-Mumford regularity of a permutation. St000653The last descent of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001778The largest greatest common divisor of an element and its image in a permutation. St000495The number of inversions of distance at most 2 of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001330The hat guessing number of a graph. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001856The number of edges in the reduced word graph of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000824The sum of the number of descents and the number of recoils of a permutation. St000527The width of the poset. St000509The diagonal index (content) of a partition. St001176The size of a partition minus its first part. St001768The number of reduced words of a signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000993The multiplicity of the largest part of an integer partition. St000259The diameter of a connected graph. St000365The number of double ascents of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000478Another weight of a partition according to Alladi. St001498The normalised height of a Nakayama algebra with magnitude 1. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001435The number of missing boxes in the first row. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001487The number of inner corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001488The number of corners of a skew partition. St001623The number of doubly irreducible elements of a lattice. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000068The number of minimal elements in a poset. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000707The product of the factorials of the parts. St000929The constant term of the character polynomial of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001568The smallest positive integer that does not appear twice in the partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000219The number of occurrences of the pattern 231 in a permutation. St000225Difference between largest and smallest parts in a partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000567The sum of the products of all pairs of parts. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001141The number of occurrences of hills of size 3 in a Dyck path. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001470The cyclic holeyness of a permutation. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001850The number of Hecke atoms of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000023The number of inner peaks of a permutation. St000045The number of linear extensions of a binary tree. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000234The number of global ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000646The number of big ascents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000663The number of right floats of a permutation. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000729The minimal arc length of a set partition. St000735The last entry on the main diagonal of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001344The neighbouring number of a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001388The number of non-attacking neighbors of a permutation. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001469The holeyness of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001712The number of natural descents of a standard Young tableau. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000444The length of the maximal rise of a Dyck path. St000502The number of successions of a set partitions. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001928The number of non-overlapping descents in a permutation. St000080The rank of the poset. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000570The Edelman-Greene number of a permutation. St000572The dimension exponent of a set partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001298The number of repeated entries in the Lehmer code of a permutation. St001424The number of distinct squares in a binary word. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice. St001626The number of maximal proper sublattices of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St000519The largest length of a factor maximising the subword complexity. St000528The height of a poset. St000677The standardized bi-alternating inversion number of a permutation. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001375The pancake length of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001705The number of occurrences of the pattern 2413 in a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000304The load of a permutation. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001077The prefix exchange distance of a permutation. St001718The number of non-empty open intervals in a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001848The atomic length of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000728The dimension of a set partition. St000008The major index of the composition. St000154The sum of the descent bottoms of a permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000305The inverse major index of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001671Haglund's hag of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000391The sum of the positions of the ones in a binary word. St001684The reduced word complexity of a permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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