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Your data matches 1140 different statistics following compositions of up to 3 maps.
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Matching statistic: St000003
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St001780
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St000057
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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
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
(load all 34 compositions to match this statistic)
(load all 34 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
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St001595
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St001929
(load all 14 compositions to match this statistic)
(load all 14 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
Description
The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path.
Matching statistic: St000034
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000175
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1]
=> [1,1]
=> 0
[1,1]
=> [2]
=> [2]
=> 0
[3]
=> [2,1]
=> [3]
=> 0
[2,1]
=> [2,1]
=> [3]
=> 0
[1,1,1]
=> [3]
=> [2,1]
=> 1
[4]
=> [3,1]
=> [2,1,1]
=> 2
[3,1]
=> [2,2]
=> [4]
=> 0
[2,2]
=> [2,1,1]
=> [3,1]
=> 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,1,1,1]
=> [4]
=> [2,2]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000290
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
St000290: 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 => 1
[2,2]
=> [[1,2],[3,4]]
=> 010 => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 0
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000293
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
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
Description
The number of inversions of a binary word.
The following 1130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000538The number of even inversions of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000747A variant of the major index of a set partition. St000836The number of descents of distance 2 of a permutation. St000849The number of 1/3-balanced pairs in a poset. St000921The number of internal inversions of a binary word. St000961The shifted major index of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive 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. 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". St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001557The number of inversions of the second entry of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001856The number of edges in the reduced word graph of a permutation. St000001The number of reduced words for a permutation. St000100The number of linear extensions of a poset. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000525The number of posets with the same zeta polynomial. St000626The minimal period of a binary word. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000988The orbit size of a permutation under Foata's bijection. 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. St001246The maximal difference between two consecutive entries of a permutation. St001268The size of the largest ordinal summand in the poset. St001313The number of Dyck paths above the lattice path given by a binary word. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001779The order of promotion on the set of linear extensions of a poset. St000064The number of one-box pattern of a permutation. St000507The number of ascents of a standard tableau. St000886The number of permutations with the same antidiagonal sums. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001462The number of factors of a standard tableaux under concatenation. St001486The number of corners of the ribbon associated with an integer composition. St000002The number of occurrences of the pattern 123 in a permutation. St000004The major index of a permutation. St000008The major index of the composition. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000081The number of edges of a graph. St000089The absolute variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000171The degree of the graph. St000174The flush statistic of a semistandard tableau. St000177The number of free tiles in the pattern. St000185The weighted size of a partition. St000204The number of internal nodes of a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. 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. St000223The number of nestings in the permutation. St000271The chromatic index of a graph. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. 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. St000429The number of occurrences of the pattern 123 or of the pattern 321 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. St000446The disorder of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000534The number of 2-rises of a permutation. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent 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. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. 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. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (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. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (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. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000663The number of right floats of a permutation. St000670The reversal length of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a 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. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000879The number of long braid edges in the graph of braid moves of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000963The 2-shifted major index of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001306The number of induced paths on four vertices in a graph. St001323The independence gap of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001341The number of edges in the center of a graph. St001375The pancake length of a permutation. St001377The major index minus the number of inversions of a permutation. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001388The number of non-attacking neighbors of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001520The number of strict 3-descents. St001535The number of cyclic alignments of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. 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. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. 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. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001644The dimension of a graph. 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. St001671Haglund's hag of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001841The number of inversions of a set partition. St001842The major index 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000074The number of special entries. St000078The number of alternating sign matrices whose left key is the permutation. St000086The number of subgraphs. St000157The number of descents of a standard tableau. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000245The number of ascents of a permutation. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. 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. St000451The length of the longest pattern of the form k 1 2. 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. St000539The number of odd inversions of a permutation. St000618The number of self-evacuating tableaux of given shape. St000619The number of cyclic descents of a permutation. St000638The number of up-down runs of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000722The number of different neighbourhoods in a graph. St000740The last entry of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000767The number of runs in an integer composition. St000789The number of crossing-similar perfect matchings of a perfect matching. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000820The number of compositions obtained by rotating the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000935The number of ordered refinements of an integer partition. St000956The maximal displacement of a permutation. St000983The length of the longest alternating subword. St000989The number of final rises of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001052The length of the exterior of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001081The number of minimal length factorizations of a permutation into star transpositions. St001093The detour number of a graph. St001096The size of the overlap set of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001267The length of the Lyndon factorization of the binary word. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001437The flex of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001806The upper middle entry of a permutation. St001883The mutual visibility number 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. St001918The degree of the cyclic sieving polynomial corresponding to an integer 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. St000062The length of the longest increasing subsequence of the permutation. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000203The number of external nodes of a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000308The height of the tree associated to a permutation. St000378The diagonal inversion number of an integer partition. St000439The position of the first down step of a Dyck path. St000485The length of the longest cycle of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000504The cardinality of the first block of a set partition. St000519The largest length of a factor maximising the subword complexity. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000823The number of unsplittable factors of the set partition. St000922The minimal number such that all substrings of this length are unique. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001062The maximal size of a block of a set partition. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001279The sum of the parts of an integer partition that are at least two. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001346The number of parking functions that give the same permutation. St001389The number of partitions of the same length below the given integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001488The number of corners of a skew partition. St001497The position of the largest weak excedence of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001746The coalition number of a graph. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000452The number of distinct eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000824The sum of the number of descents and the number of recoils of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. 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. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001964The interval resolution global dimension of a poset. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000012The area of a Dyck path. St000145The Dyson rank of a partition. St000216The absolute length of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000225Difference between largest and smallest parts in a partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000454The largest eigenvalue of a graph if it is integral. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000794The mak of a permutation. St000809The reduced reflection length of the permutation. St000850The number of 1/2-balanced pairs in a poset. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000984The number of boxes below precisely one peak. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001252Half the sum of the even parts of a partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001822The number of alignments of a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000045The number of linear extensions of a binary tree. St000633The size of the automorphism group of a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000741The Colin de Verdière graph invariant. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001399The distinguishing number of a poset. St001470The cyclic holeyness of a permutation. St001487The number of inner corners of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001768The number of reduced words of a signed permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000297The number of leading ones in a binary word. St000360The number of occurrences of the pattern 32-1. St000392The length of the longest run of ones in a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St000982The length of the longest constant subword. St001330The hat guessing number of a graph. St001424The number of distinct squares in a binary word. St001530The depth of a Dyck path. St000141The maximum drop size of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001127The sum of the squares of the parts of a partition. St001161The major index north count of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001541The Gini index of an integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000527The width of the poset. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000009The charge of a standard tableau. St000013The height of a Dyck path. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000041The number of nestings of a perfect matching. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000091The descent variation of a composition. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000169The cocharge of a standard tableau. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000210Minimum over maximum difference of elements in cycles. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000288The number of ones in a binary word. St000304The load of a permutation. St000305The inverse major index of a permutation. St000306The bounce count of a Dyck path. St000330The (standard) major index of a standard tableau. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000408The number of occurrences of the pattern 4231 in a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000616The inversion index of a permutation. St000674The number of hills of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000753The Grundy value for the game of Kayles on a binary word. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000828The spearman's rho of a permutation and the identity permutation. St000833The comajor index of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000979Half of MacMahon's equal index of a Dyck path. St000992The alternating sum of the parts of an integer partition. St000995The largest even part of an integer partition. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001248Sum of the even parts of a partition. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001379The number of inversions plus the major index of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001586The number of odd parts smaller than the largest even part in an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001684The reduced word complexity of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001726The number of visible inversions of a permutation. St001777The number of weak descents in an integer composition. St001846The number of elements which do not have a complement in the lattice. St001956The comajor index for set-valued two-row standard Young tableaux. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000842The breadth of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000090The variation of a composition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000233The number of nestings of a set partition. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001615The number of join prime elements of a lattice. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St000254The nesting number of a set partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001490The number of connected components of a skew partition. St001820The size of the image of the pop stack sorting operator. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001741The largest integer such that all patterns of this size are contained in the permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000936The number of even values of the symmetric group character corresponding to the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000068The number of minimal elements in a poset. St001625The Möbius invariant of a lattice. St000679The pruning number of an ordered tree. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St001875The number of simple modules with projective dimension at most 1. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St000420The number of Dyck paths that are weakly above a Dyck path. St000567The sum of the products of all pairs of parts. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St001808The box weight or horizontal decoration of a Dyck path. St000022The number of fixed points of a permutation. St000082The number of elements smaller than a binary tree in Tamari order. St000083The number of left oriented leafs of a binary tree except the first one. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000259The diameter of a connected graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000461The rix statistic of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000490The intertwining number of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000624The normalized sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000673The number of non-fixed points of a permutation. St000675The number of centered multitunnels of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000728The dimension of a set partition. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000829The Ulam distance of a permutation to the identity permutation. St000830The total displacement of a permutation. St000831The number of indices that are either descents or recoils. St000844The size of the largest block in the direct sum decomposition of a permutation. St000873The aix statistic of a permutation. St000874The position of the last double rise in a Dyck path. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000925The number of topologically connected components of a set partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000990The first ascent of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001118The acyclic chromatic index of a graph. St001141The number of occurrences of hills of size 3 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001371The length of the longest Yamanouchi prefix of a binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001500The global dimension of magnitude 1 Nakayama algebras. St001524The degree of symmetry of a binary word. St001531Number of partial orders contained in the poset determined by the Dyck path. St001721The degree of a binary word. St001731The factorization defect of a permutation. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001930The weak major index of a binary word. St001959The product of the heights of the peaks of a Dyck path. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000862The number of parts of the shifted shape of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000782The indicator function of whether a given perfect matching is an L & P matching. 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 $K[x]/(x^n)$. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000048The multinomial of the parts of a partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000159The number of distinct parts of the integer partition. St000179The product of the hook lengths of the integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000296The length of the symmetric border of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000346The number of coarsenings of a partition. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000443The number of long tunnels of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. 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. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 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. St000623The number of occurrences of the pattern 52341 in a permutation. St000627The exponent of a binary word. St000628The balance of a binary word. St000629The defect of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000649The number of 3-excedences of a permutation. St000655The length of the minimal rise of a Dyck path. St000666The number of right tethers of a permutation. St000667The greatest common divisor of the parts of the partition. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000783The side length of the largest staircase partition fitting into a partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000869The sum of the hook lengths of an integer partition. St000872The number of very big descents of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000943The number of spots the most unlucky car had to go further in a parking function. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001128The exponens consonantiae of a partition. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001381The fertility of a permutation. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001396Number of triples of incomparable elements in a finite poset. St001400The total number of Littlewood-Richardson tableaux of given shape. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001481The minimal height of a peak of a Dyck path. St001484The number of singletons of an integer partition. St001513The number of nested exceedences of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001561The value of the elementary symmetric function evaluated at 1. St001571The Cartan determinant of the integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001712The number of natural descents of a standard Young tableau. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001732The number of peaks visible from the left. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000037The sign of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000694The number of affine bounded permutations that project to a given permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001469The holeyness of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000168The number of internal nodes of an ordered tree. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000487The length of the shortest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. 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. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001075The minimal size of a block of a set partition. 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 $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001569The maximal modular displacement of a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St000080The rank of the poset. St000136The dinv of a parking function. St000166The depth minus 1 of an ordered tree. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000213The number of weak exceedances (also weak excedences) of a permutation. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001405The number of bonds in a permutation. St001517The length of a longest pair of twins in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St001667The maximal size of a pair of weak twins for a permutation. St001754The number of tolerances of a finite lattice. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001874Lusztig's a-function for the symmetric group. St001926Sparre Andersen's position of the maximum of a signed permutation. St000116The major index of a semistandard tableau obtained by standardizing. St000135The number of lucky cars of the parking function. St000528The height of a poset. St000863The length of the first row of the shifted shape of a permutation. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001645The pebbling number of a connected graph. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000492The rob 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. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000912The number of maximal antichains in a poset. St001160The number of proper blocks (or intervals) of a permutations. St001343The dimension of the reduced incidence algebra of a poset. 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. St000501The size of the first part in the decomposition of a permutation. St000540The sum of the entries of a parking function minus its length. St000756The sum of the positions of the left to right maxima of a permutation. St001519The pinnacle sum of a permutation. St001536The number of cyclic misalignments of a permutation. St001807The lower middle entry of a permutation. St001468The smallest fixpoint of a permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000543The size of the conjugacy class of a binary word. St000625The sum of the minimal distances to a greater element. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000165The sum of the entries of a parking function. St001858The number of covering elements of a signed permutation in absolute order. St001865The number of alignments of a signed permutation. St000391The sum of the positions of the ones in a binary word. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St000690The size of the conjugacy class of a permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000180The number of chains of a poset. St001852The size of the conjugacy class of the signed permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001885The number of binary words with the same proper border set. St000324The shape of the tree associated to a permutation. St000529The number of permutations whose descent word is the given binary word. St001528The number of permutations such that the product with the permutation has the same number of fixed points.
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