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Your data matches 340 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[]],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[[[]],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 8
[]
=> []
=> ([],1)
=> 0
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000005
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 6 = 8 - 2
[]
=> ?
=> ?
=> ? = 0 - 2
Description
The bounce statistic of a Dyck path.
The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$.
The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$.
In particular, the bounce statistics of $D$ and $D'$ coincide.
Matching statistic: St001389
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,2]
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> 5 = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => [4,2]
=> 7 = 8 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [4,2]
=> 7 = 8 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => [4,2]
=> 7 = 8 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => [4,2]
=> 7 = 8 - 1
[]
=> []
=> [] => []
=> ? = 0 - 1
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000012
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 6 = 8 - 2
[]
=> ?
=> ?
=> ?
=> ? = 0 - 2
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000029
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,3,2] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,3,2] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,5,6,3,4,2] => 6 = 8 - 2
[]
=> ?
=> ? => ? => ? = 0 - 2
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: St000133
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 6 = 8 - 2
[]
=> ?
=> ? => ? => ? = 0 - 2
Description
The "bounce" of a permutation.
Matching statistic: St000218
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 6 = 8 - 2
[]
=> ?
=> ?
=> ? => ? = 0 - 2
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000222
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [5,4,3,2,1,6] => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [6,5,3,4,2,1] => 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [5,4,3,2,1,6] => 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [6,5,3,4,2,1] => 6 = 8 - 2
[]
=> []
=> [] => [] => ? = 0 - 2
Description
The number of alignments in the permutation.
Matching statistic: St000223
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 6 = 8 - 2
[]
=> []
=> ?
=> ? => ? = 0 - 2
Description
The number of nestings in the permutation.
Matching statistic: St000304
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => 6 = 8 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => 6 = 8 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => 6 = 8 - 2
[]
=> []
=> ?
=> ? => ? = 0 - 2
Description
The load of a permutation.
The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.
The following 330 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000355The number of occurrences of the pattern 21-3. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001161The major index north count of a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001349The number of different graphs obtained from the given graph by removing an edge. 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. St001727The number of invisible inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001874Lusztig's a-function for the symmetric group. St000438The position of the last up step in a Dyck path. 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. St000538The number of even inversions of a permutation. St000641The number of non-empty boolean intervals in a poset. St000824The sum of the number of descents and the number of recoils of a permutation. St000625The sum of the minimal distances to a greater element. St000708The product of the parts of an integer partition. St000795The mad of a permutation. St000796The stat' of a permutation. St000830The total displacement of a permutation. St000874The position of the last double rise in a Dyck path. St001721The degree of a binary word. St000082The number of elements smaller than a binary tree in Tamari order. St000376The bounce deficit of a Dyck path. St000477The weight of a partition according to Alladi. St000539The number of odd inversions of a permutation. St000673The number of non-fixed points of a permutation. St000947The major index east count of a Dyck path. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000520The number of patterns in a permutation. St000941The number of characters of the symmetric group whose value on the partition is even. St001246The maximal difference between two consecutive entries of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000836The number of descents of distance 2 of a permutation. St000956The maximal displacement of a permutation. St000981The length of the longest zigzag subpath. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000311The number of vertices of odd degree in a graph. St000235The number of indices that are not cyclical small weak excedances. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001565The number of arithmetic progressions of length 2 in a permutation. St001703The villainy of a graph. St000652The maximal difference between successive positions of a permutation. St000064The number of one-box pattern of a permutation. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001486The number of corners of the ribbon associated with an integer composition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000494The number of inversions of distance at most 3 of a permutation. St000653The last descent of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001497The position of the largest weak excedence of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (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. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000039The number of crossings of a permutation. St000216The absolute length of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000831The number of indices that are either descents or recoils. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001298The number of repeated entries in the Lehmer code of a permutation. St001432The order dimension of the partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000997The even-odd crank of an integer partition. St001569The maximal modular displacement of a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000327The number of cover relations in a poset. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001811The Castelnuovo-Mumford regularity of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000806The semiperimeter of the associated bargraph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000921The number of internal inversions of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000885The number of critical steps in the Catalan decomposition of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. 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 $eA$. St001892The flag excedance statistic of a signed permutation. St001894The depth of a signed permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000936The number of even values of the symmetric group character corresponding to the partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001375The pancake length of a permutation. St001817The number of flag weak exceedances of a signed permutation. 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. St000238The number of indices that are not small weak excedances. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001117The game chromatic index of a graph. St001488The number of corners of a skew partition. St001806The upper middle entry of a permutation. St001875The number of simple modules with projective dimension at most 1. St000392The length of the longest run of ones in a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000770The major index of an integer partition when read from bottom to top. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000667The greatest common divisor of the parts of the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001383The BG-rank of an integer 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000390The number of runs of ones in a binary word. St001267The length of the Lyndon factorization of the binary word. 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. St001424The number of distinct squares in a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001621The number of atoms of a lattice. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001948The number of augmented double ascents of a permutation. St000292The number of ascents of a binary word. St000632The jump number of the poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000877The depth of the binary word interpreted as a path. 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 $K[x]/(x^n)$. St001372The length of a longest cyclic run of ones of a binary word. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001862The number of crossings of a signed permutation. St001893The flag descent of a signed permutation. St001935The number of ascents in a parking function. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000381The largest part of an integer composition. St000543The size of the conjugacy class of a binary word. St000545The number of parabolic double cosets with minimal element being the given permutation. St000626The minimal period of a binary word. St000808The number of up steps of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000891The number of distinct diagonal sums of a permutation matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000904The maximal number of repetitions of an integer composition. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. 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. St001437The flex of a binary word. St001566The length of the longest arithmetic progression in a permutation. St001626The number of maximal proper sublattices of a lattice. St001637The number of (upper) dissectors of a poset. St001733The number of weak left to right maxima of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000245The number of ascents of a permutation. St000295The length of the border of a binary word. St000451The length of the longest pattern of the form k 1 2. St000519The largest length of a factor maximising the subword complexity. St000527The width of the poset. St000630The length of the shortest palindromic decomposition of a binary word. St000635The number of strictly order preserving maps of a poset into itself. St000670The reversal length 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. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000737The last entry on the main diagonal of a semistandard tableau. St000738The first entry in the last row of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000767The number of runs in an integer composition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000820The number of compositions obtained by rotating the composition. St000834The number of right outer peaks of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000899The maximal number of repetitions of an integer composition. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000942The number of critical left to right maxima of the parking functions. 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. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001268The size of the largest ordinal summand in the poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001399The distinguishing number of a poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001510The number of self-evacuating linear extensions of a finite poset. St001524The degree of symmetry of a binary word. St001533The largest coefficient of the Poincare polynomial of the poset cone. 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. St001668The number of points of the poset minus the width of the poset. St001769The reflection length of a signed permutation. St001779The order of promotion on the set of linear extensions of a poset. St001819The flag Denert index of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001863The number of weak excedances of a signed permutation. St001884The number of borders of a binary word. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000017The number of inversions of a standard tableau. St000028The number of stack-sorts needed to sort a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000077The number of boxed and circled entries. St000136The dinv of a parking function. St000141The maximum drop size of a permutation. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000359The number of occurrences of the pattern 23-1. 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. St000389The number of runs of ones of odd length in a binary word. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000628The balance of a binary word. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001209The pmaj statistic of a parking function. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001433The flag major index of a signed permutation. St001485The modular major index of a binary word. St001545The second Elser number of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001777The number of weak descents in an integer composition. St001821The sorting index of a signed permutation. St001848The atomic length of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001946The number of descents in a parking function. St000187The determinant of an alternating sign matrix. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001541The Gini index of an integer partition. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001763The Hurwitz number of an integer partition. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition.
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