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Your data matches 610 different statistics following compositions of up to 3 maps.
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Matching statistic: St000004
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[1,2,3,4] => 0
[1,3,2,4] => 2
[1,4,2,3] => 2
[2,1,3,4] => 1
[2,3,1,4] => 2
[2,4,1,3] => 2
[3,1,2,4] => 1
[3,4,1,2] => 2
[4,1,2,3] => 1
[1,2,3,4,5] => 0
[2,1,3,4,5] => 1
[3,1,2,4,5] => 1
[4,1,2,3,5] => 1
[5,1,2,3,4] => 1
[1,2,3,4,5,6] => 0
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000653
(load all 69 compositions to match this statistic)
(load all 69 compositions to match this statistic)
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[1,2,3,4] => 0
[1,3,2,4] => 2
[1,4,2,3] => 2
[2,1,3,4] => 1
[2,3,1,4] => 2
[2,4,1,3] => 2
[3,1,2,4] => 1
[3,4,1,2] => 2
[4,1,2,3] => 1
[1,2,3,4,5] => 0
[2,1,3,4,5] => 1
[3,1,2,4,5] => 1
[4,1,2,3,5] => 1
[5,1,2,3,4] => 1
[1,2,3,4,5,6] => 0
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000794
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(load all 28 compositions to match this statistic)
St000794: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[1,2,3,4] => 0
[1,3,2,4] => 2
[1,4,2,3] => 2
[2,1,3,4] => 1
[2,3,1,4] => 2
[2,4,1,3] => 2
[3,1,2,4] => 1
[3,4,1,2] => 2
[4,1,2,3] => 1
[1,2,3,4,5] => 0
[2,1,3,4,5] => 1
[3,1,2,4,5] => 1
[4,1,2,3,5] => 1
[5,1,2,3,4] => 1
[1,2,3,4,5,6] => 0
Description
The mak of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(2\underline{31})$, $(\underline{32}1)$, $(1\underline{32})$, $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000008
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => 0
[2,1] => [1,1] => 1
[1,2,3] => [3] => 0
[1,3,2] => [2,1] => 2
[2,1,3] => [1,2] => 1
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[1,2,3,4] => [4] => 0
[1,3,2,4] => [2,2] => 2
[1,4,2,3] => [2,2] => 2
[2,1,3,4] => [1,3] => 1
[2,3,1,4] => [2,2] => 2
[2,4,1,3] => [2,2] => 2
[3,1,2,4] => [1,3] => 1
[3,4,1,2] => [2,2] => 2
[4,1,2,3] => [1,3] => 1
[1,2,3,4,5] => [5] => 0
[2,1,3,4,5] => [1,4] => 1
[3,1,2,4,5] => [1,4] => 1
[4,1,2,3,5] => [1,4] => 1
[5,1,2,3,4] => [1,4] => 1
[1,2,3,4,5,6] => [6] => 0
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000018
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => 2
[2,4,1,3] => [2,1,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [1,3,4,2] => 2
[4,1,2,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [2,1,3,4,5] => 1
[3,1,2,4,5] => [1,3,2,4,5] => 1
[4,1,2,3,5] => [1,2,4,3,5] => 1
[5,1,2,3,4] => [1,2,3,5,4] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000019
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => 2
[2,4,1,3] => [2,1,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [1,3,4,2] => 2
[4,1,2,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [2,1,3,4,5] => 1
[3,1,2,4,5] => [1,3,2,4,5] => 1
[4,1,2,3,5] => [1,2,4,3,5] => 1
[5,1,2,3,4] => [1,2,3,5,4] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000029
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => 2
[2,4,1,3] => [2,1,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [1,3,4,2] => 2
[4,1,2,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [2,1,3,4,5] => 1
[3,1,2,4,5] => [1,3,2,4,5] => 1
[4,1,2,3,5] => [1,2,4,3,5] => 1
[5,1,2,3,4] => [1,2,3,5,4] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The depth of a permutation.
This is given by
$$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$
The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$.
Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000030
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => 2
[2,4,1,3] => [2,1,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [1,3,4,2] => 2
[4,1,2,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [2,1,3,4,5] => 1
[3,1,2,4,5] => [1,3,2,4,5] => 1
[4,1,2,3,5] => [1,2,4,3,5] => 1
[5,1,2,3,4] => [1,2,3,5,4] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000156
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 2
[3,1,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [3,2,1,4] => 2
[2,4,1,3] => [4,2,1,3] => 2
[3,1,2,4] => [3,1,2,4] => 1
[3,4,1,2] => [4,1,3,2] => 2
[4,1,2,3] => [4,1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [2,1,3,4,5] => 1
[3,1,2,4,5] => [3,1,2,4,5] => 1
[4,1,2,3,5] => [4,1,2,3,5] => 1
[5,1,2,3,4] => [5,1,2,3,4] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The Denert index of a permutation.
It is defined as
$$
\begin{align*}
den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\
&+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\
&+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \}
\end{align*}
$$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].
Matching statistic: St000216
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 2
[3,1,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,3,1,4] => [2,3,1,4] => 2
[2,4,1,3] => [2,1,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [1,3,4,2] => 2
[4,1,2,3] => [1,2,4,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [2,1,3,4,5] => 1
[3,1,2,4,5] => [1,3,2,4,5] => 1
[4,1,2,3,5] => [1,2,4,3,5] => 1
[5,1,2,3,4] => [1,2,3,5,4] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
The following 600 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000305The inverse major index of a permutation. St000330The (standard) major index of a standard tableau. St000391The sum of the positions of the ones in a binary word. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000670The reversal length of a permutation. St000693The modular (standard) major index of a standard tableau. St000792The Grundy value for the game of ruler on a binary word. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001090The number of pop-stack-sorts needed to sort a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000638The number of up-down runs of a permutation. St000738The first entry in the last row of a standard tableau. St000005The bounce statistic of a Dyck path. St000009The charge of a standard tableau. St000021The number of descents of a permutation. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000271The chromatic index of a graph. St000289The decimal representation of a binary word. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000446The disorder of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000502The number of successions of a set partitions. St000539The number of odd inversions of a permutation. St000651The maximal size of a rise in a permutation. St000703The number of deficiencies of a permutation. St000728The dimension of a set partition. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000849The number of 1/3-balanced pairs in a poset. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001161The major index north count of a Dyck path. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001479The number of bridges of a graph. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001649The length of a longest trail in a graph. St001671Haglund's hag of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001721The degree of a binary word. St001726The number of visible inversions of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St000054The first entry of the permutation. St000058The order of a permutation. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000345The number of refinements of a partition. 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. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000839The largest opener of a set partition. St000935The number of ordered refinements of an integer partition. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001497The position of the largest weak excedence of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. 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. St000006The dinv of a Dyck path. St000010The length of the partition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000051The size of the left subtree of a binary tree. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000142The number of even parts of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000154The sum of the descent bottoms of a permutation. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000161The sum of the sizes of the right subtrees of a binary tree. St000185The weighted size of a partition. St000228The size of a partition. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000472The sum of the ascent bottoms of a permutation. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000490The intertwining number of a set partition. St000493The los statistic of a set partition. St000503The maximal difference between two elements in a common block. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000548The number of different non-empty partial sums of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000730The maximal arc length of a set partition. St000742The number of big ascents of a permutation after prepending zero. St000753The Grundy value for the game of Kayles on a binary word. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000845The maximal number of elements covered by an element in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000984The number of boxes below precisely one peak. St000992The alternating sum of the parts of an integer partition. St001033The normalized 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. St001077The prefix exchange distance of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001127The sum of the squares of the parts of a partition. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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 St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001341The number of edges in the center of a graph. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001375The pancake length of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001427The number of descents of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001584The area statistic between a Dyck path and its bounce path. St001592The maximal number of simple paths between any two different vertices of a graph. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St001622The number of join-irreducible elements of a lattice. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001743The discrepancy of a graph. St001759The Rajchgot index of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001792The arboricity of a graph. St001812The biclique partition number 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. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000082The number of elements smaller than a binary tree in Tamari order. St000088The row sums of the character table of the symmetric group. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000240The number of indices that are not small excedances. St000300The number of independent sets of vertices of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000420The number of Dyck paths that are weakly above a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000507The number of ascents of a standard tableau. St000532The total number of rook placements on a Ferrers board. St000542The number of left-to-right-minima of a permutation. 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. St000722The number of different neighbourhoods in a graph. St000739The first entry in the last row of a semistandard tableau. St000740The last entry of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. 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. St000822The Hadwiger number of the graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000883The number of longest increasing subsequences of a permutation. St000925The number of topologically connected components of a set partition. St000933The number of multipartitions of sizes given by an integer partition. St000971The smallest closer of a set partition. St000983The length of the longest alternating subword. St000990The first ascent of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001062The maximal size of a block of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001285The number of primes in the column sums of the two line notation of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001330The hat guessing number of a graph. St001346The number of parking functions that give the same permutation. 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. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001808The box weight or horizontal decoration of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000439The position of the first down step of a Dyck path. 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. St000095The number of triangles of a graph. St000327The number of cover relations in a poset. St000840The number of closers smaller than the largest opener in a perfect matching. St000873The aix statistic of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001742The difference of the maximal and the minimal degree in a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St001555The order of a signed permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001118The acyclic chromatic index of a graph. St000087The number of induced subgraphs. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000343The number of spanning subgraphs of a graph. St000364The exponent of the automorphism group of a graph. St000452The number of distinct eigenvalues of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000636The hull number of a graph. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000972The composition number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. 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. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. 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. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001802The number of endomorphisms of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001892The flag excedance statistic of a signed permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000145The Dyson rank of a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000361The second Zagreb index of a graph. St000387The matching number of a graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000928The sum of the coefficients of the character polynomial of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001827The number of two-component spanning forests of a graph. St001949The rigidity index of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001931The weak major index of an integer composition regarded as a word. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000770The major index of an integer partition when read from bottom to top. St001209The pmaj statistic of a parking function. St001433The flag major index of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001060The distinguishing index of a graph. 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 $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000456The monochromatic index of a connected graph. St000769The major index of a composition regarded as a word. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. 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. 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. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000464The Schultz index of a connected graph. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. 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. St001545The second Elser number of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000781The number of proper colouring schemes of a Ferrers diagram. St000806The semiperimeter of the associated bargraph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001281The normalized isoperimetric number of a graph. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the 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. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001875The number of simple modules with projective dimension at most 1. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001926Sparre Andersen's position of the maximum of a signed permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000474Dyson's crank of a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001651The Frankl number of a lattice. 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 $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. 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. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000699The toughness times the least common multiple of 1,.
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