Your data matches 230 different statistics following compositions of up to 3 maps.
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Matching statistic: St000904
(load all 4 compositions to match this statistic)
Mapping
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 2 - 1
[1,1] => 2 = 3 - 1
[2] => 1 = 2 - 1
[1,1,1] => 3 = 4 - 1
[1,2] => 1 = 2 - 1
[2,1] => 1 = 2 - 1
[3] => 1 = 2 - 1
[1,1,1,1] => 4 = 5 - 1
[1,1,2] => 2 = 3 - 1
[1,2,1] => 2 = 3 - 1
[1,3] => 1 = 2 - 1
[2,1,1] => 2 = 3 - 1
[2,2] => 2 = 3 - 1
[3,1] => 1 = 2 - 1
[4] => 1 = 2 - 1
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St000160
(load all 4 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 1 = 2 - 1
[1,1] => [1,1]
 => 2 = 3 - 1
[2] => [2]
 => 1 = 2 - 1
[1,1,1] => [1,1,1]
 => 3 = 4 - 1
[1,2] => [2,1]
 => 1 = 2 - 1
[2,1] => [2,1]
 => 1 = 2 - 1
[3] => [3]
 => 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
 => 4 = 5 - 1
[1,1,2] => [2,1,1]
 => 2 = 3 - 1
[1,2,1] => [2,1,1]
 => 2 = 3 - 1
[1,3] => [3,1]
 => 1 = 2 - 1
[2,1,1] => [2,1,1]
 => 2 = 3 - 1
[2,2] => [2,2]
 => 2 = 3 - 1
[3,1] => [3,1]
 => 1 = 2 - 1
[4] => [4]
 => 1 = 2 - 1
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St001933
(load all 4 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 1 = 2 - 1
[1,1] => [1,1]
 => 2 = 3 - 1
[2] => [2]
 => 1 = 2 - 1
[1,1,1] => [1,1,1]
 => 3 = 4 - 1
[1,2] => [2,1]
 => 1 = 2 - 1
[2,1] => [2,1]
 => 1 = 2 - 1
[3] => [3]
 => 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
 => 4 = 5 - 1
[1,1,2] => [2,1,1]
 => 2 = 3 - 1
[1,2,1] => [2,1,1]
 => 2 = 3 - 1
[1,3] => [3,1]
 => 1 = 2 - 1
[2,1,1] => [2,1,1]
 => 2 = 3 - 1
[2,2] => [2,2]
 => 2 = 3 - 1
[3,1] => [3,1]
 => 1 = 2 - 1
[4] => [4]
 => 1 = 2 - 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001091
(load all 4 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 0 = 2 - 2
[1,1] => [1,1]
 => 1 = 3 - 2
[2] => [2]
 => 0 = 2 - 2
[1,1,1] => [1,1,1]
 => 2 = 4 - 2
[1,2] => [2,1]
 => 0 = 2 - 2
[2,1] => [2,1]
 => 0 = 2 - 2
[3] => [3]
 => 0 = 2 - 2
[1,1,1,1] => [1,1,1,1]
 => 3 = 5 - 2
[1,1,2] => [2,1,1]
 => 1 = 3 - 2
[1,2,1] => [2,1,1]
 => 1 = 3 - 2
[1,3] => [3,1]
 => 0 = 2 - 2
[2,1,1] => [2,1,1]
 => 1 = 3 - 2
[2,2] => [2,2]
 => 1 = 3 - 2
[3,1] => [3,1]
 => 0 = 2 - 2
[4] => [4]
 => 0 = 2 - 2
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001180
(load all 3 compositions to match this statistic)
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001180: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 2
[1,1] => [2] => [1,1,0,0]
 => 3
[2] => [1] => [1,0]
 => 2
[1,1,1] => [3] => [1,1,1,0,0,0]
 => 4
[1,2] => [1,1] => [1,0,1,0]
 => 2
[2,1] => [1,1] => [1,0,1,0]
 => 2
[3] => [1] => [1,0]
 => 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 5
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,3] => [1,1] => [1,0,1,0]
 => 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,2] => [2] => [1,1,0,0]
 => 3
[3,1] => [1,1] => [1,0,1,0]
 => 2
[4] => [1] => [1,0]
 => 2
Description
Number of indecomposable injective modules with projective dimension at most 1.
Matching statistic: St001211
(load all 6 compositions to match this statistic)
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 2
[1,1] => [2] => [1,1,0,0]
 => 3
[2] => [1] => [1,0]
 => 2
[1,1,1] => [3] => [1,1,1,0,0,0]
 => 4
[1,2] => [1,1] => [1,0,1,0]
 => 2
[2,1] => [1,1] => [1,0,1,0]
 => 2
[3] => [1] => [1,0]
 => 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 5
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,3] => [1,1] => [1,0,1,0]
 => 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,2] => [2] => [1,1,0,0]
 => 3
[3,1] => [1,1] => [1,0,1,0]
 => 2
[4] => [1] => [1,0]
 => 2
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 5 compositions to match this statistic)
Mapping
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 2
[1,1] => [2] => [1,1,0,0]
 => 3
[2] => [1] => [1,0]
 => 2
[1,1,1] => [3] => [1,1,1,0,0,0]
 => 4
[1,2] => [1,1] => [1,0,1,0]
 => 2
[2,1] => [1,1] => [1,0,1,0]
 => 2
[3] => [1] => [1,0]
 => 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 5
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 3
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,3] => [1,1] => [1,0,1,0]
 => 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 3
[2,2] => [2] => [1,1,0,0]
 => 3
[3,1] => [1,1] => [1,0,1,0]
 => 2
[4] => [1] => [1,0]
 => 2
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000392
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Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 10 => 1 = 2 - 1
[1,1] => [1,1]
 => 110 => 2 = 3 - 1
[2] => [2]
 => 100 => 1 = 2 - 1
[1,1,1] => [1,1,1]
 => 1110 => 3 = 4 - 1
[1,2] => [2,1]
 => 1010 => 1 = 2 - 1
[2,1] => [2,1]
 => 1010 => 1 = 2 - 1
[3] => [3]
 => 1000 => 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
 => 11110 => 4 = 5 - 1
[1,1,2] => [2,1,1]
 => 10110 => 2 = 3 - 1
[1,2,1] => [2,1,1]
 => 10110 => 2 = 3 - 1
[1,3] => [3,1]
 => 10010 => 1 = 2 - 1
[2,1,1] => [2,1,1]
 => 10110 => 2 = 3 - 1
[2,2] => [2,2]
 => 1100 => 2 = 3 - 1
[3,1] => [3,1]
 => 10010 => 1 = 2 - 1
[4] => [4]
 => 10000 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000876
(load all 6 compositions to match this statistic)
Mapping
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 => 1 = 2 - 1
[1,1] => 11 => 11 => 2 = 3 - 1
[2] => 10 => 10 => 1 = 2 - 1
[1,1,1] => 111 => 111 => 3 = 4 - 1
[1,2] => 110 => 110 => 1 = 2 - 1
[2,1] => 101 => 110 => 1 = 2 - 1
[3] => 100 => 100 => 1 = 2 - 1
[1,1,1,1] => 1111 => 1111 => 4 = 5 - 1
[1,1,2] => 1110 => 1110 => 2 = 3 - 1
[1,2,1] => 1101 => 1110 => 2 = 3 - 1
[1,3] => 1100 => 1010 => 1 = 2 - 1
[2,1,1] => 1011 => 1110 => 2 = 3 - 1
[2,2] => 1010 => 1100 => 1 = 2 - 1
[3,1] => 1001 => 1010 => 1 = 2 - 1
[4] => 1000 => 1000 => 2 = 3 - 1
Description
The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000907
(load all 4 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => ([],1)
 => 1 = 2 - 1
[1,1] => [1,0,1,0]
 => ([(0,1)],2)
 => 2 = 3 - 1
[2] => [1,1,0,0]
 => ([],2)
 => 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,2] => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
 => ([],3)
 => 1 = 2 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 2 = 3 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 3 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 1 = 2 - 1
Description
The number of maximal antichains of minimal length in a poset.
The following 220 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001571The Cartan determinant of the integer partition. St000013The height of a Dyck path. St000144The pyramid weight of the Dyck path. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000628The balance of a binary word. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000863The length of the first row of the shifted shape of a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000062The length of the longest increasing subsequence of the permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000393The number of strictly increasing runs in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000443The number of long tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000627The exponent of a binary word. St000653The last descent of a permutation. St000654The first descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000794The mak of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000877The depth of the binary word interpreted as a path. St000899The maximal number of repetitions of an integer composition. St000910The number of maximal chains of minimal length in a poset. St000922The minimal number such that all substrings of this length are unique. St000947The major index east count of a Dyck path. St000996The number of exclusive left-to-right maxima 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. St001007Number of simple modules with projective dimension 1 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001203We associate to 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 Dyck path as follows: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001267The length of the Lyndon factorization of the binary word. St001286The annihilation number of a graph. 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. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001399The distinguishing number of a poset. St001415The length of the longest 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. St001437The flex of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. 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. St001884The number of borders of a binary word. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000245The number of ascents of a permutation. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000356The number of occurrences of the pattern 13-2. St000365The number of double ascents of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000516The number of stretching pairs of a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 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. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000672The number of minimal elements in Bruhat order not less than the permutation. St000674The number of hills of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000931The number of occurrences of the pattern UUU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000989The number of final rises of a permutation. 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. 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. 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. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001721The degree of a binary word. St001777The number of weak descents in an integer composition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001948The number of augmented double ascents of a permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000850The number of 1/2-balanced pairs in a poset. St001330The hat guessing number of a graph. St000438The position of the last up step in a Dyck path. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St000650The number of 3-rises of a permutation. St000806The semiperimeter of the associated bargraph. St000014The number of parking functions supported by a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000294The number of distinct factors of a binary word. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001012Number of simple modules with projective dimension at most 2 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. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 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. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001400The total number of Littlewood-Richardson tableaux of given shape. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001814The number of partitions interlacing the given partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001645The pebbling number of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000075The orbit size of a standard tableau under promotion. St001568The smallest positive integer that does not appear twice in the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000736The last entry in the first row of a semistandard tableau. 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 $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation.