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Your data matches 423 different statistics following compositions of up to 3 maps.
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Matching statistic: St001389
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 0 + 1
[2]
=> 2 = 1 + 1
[1,1]
=> 1 = 0 + 1
[3]
=> 3 = 2 + 1
[2,1]
=> 2 = 1 + 1
[1,1,1]
=> 1 = 0 + 1
[4]
=> 4 = 3 + 1
[3,1]
=> 3 = 2 + 1
[2,2]
=> 3 = 2 + 1
[2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 1 = 0 + 1
[5]
=> 5 = 4 + 1
[4,1]
=> 4 = 3 + 1
[3,2]
=> 5 = 4 + 1
[3,1,1]
=> 3 = 2 + 1
[2,2,1]
=> 3 = 2 + 1
[2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> 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: St000108
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 1 = 0 + 1
[2]
=> []
=> 1 = 0 + 1
[1,1]
=> [1]
=> 2 = 1 + 1
[3]
=> []
=> 1 = 0 + 1
[2,1]
=> [1]
=> 2 = 1 + 1
[1,1,1]
=> [1,1]
=> 3 = 2 + 1
[4]
=> []
=> 1 = 0 + 1
[3,1]
=> [1]
=> 2 = 1 + 1
[2,2]
=> [2]
=> 3 = 2 + 1
[2,1,1]
=> [1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,1,1]
=> 4 = 3 + 1
[5]
=> []
=> 1 = 0 + 1
[4,1]
=> [1]
=> 2 = 1 + 1
[3,2]
=> [2]
=> 3 = 2 + 1
[3,1,1]
=> [1,1]
=> 3 = 2 + 1
[2,2,1]
=> [2,1]
=> 5 = 4 + 1
[2,1,1,1]
=> [1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 5 = 4 + 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 1 = 0 + 1
[2]
=> []
=> 1 = 0 + 1
[1,1]
=> [1]
=> 2 = 1 + 1
[3]
=> []
=> 1 = 0 + 1
[2,1]
=> [1]
=> 2 = 1 + 1
[1,1,1]
=> [1,1]
=> 3 = 2 + 1
[4]
=> []
=> 1 = 0 + 1
[3,1]
=> [1]
=> 2 = 1 + 1
[2,2]
=> [2]
=> 3 = 2 + 1
[2,1,1]
=> [1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,1,1]
=> 4 = 3 + 1
[5]
=> []
=> 1 = 0 + 1
[4,1]
=> [1]
=> 2 = 1 + 1
[3,2]
=> [2]
=> 3 = 2 + 1
[3,1,1]
=> [1,1]
=> 3 = 2 + 1
[2,2,1]
=> [2,1]
=> 5 = 4 + 1
[2,1,1,1]
=> [1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 5 = 4 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000030
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
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: St000394
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000645
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St001278
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Matching statistic: St001684
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
Description
The reduced word complexity of a permutation.
For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$.
For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$.
This statistic appears in [1, Question 6.1].
Matching statistic: St000110
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5 = 4 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001464
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5 = 4 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
The following 413 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000133The "bounce" of a permutation. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000305The inverse major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000446The disorder of a permutation. St000459The hook length of the base cell of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St001034The area of the parallelogram polyomino associated with the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001161The major index north count of a Dyck path. St001485The modular major index of a binary word. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001759The Rajchgot index of a permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000005The bounce statistic of a Dyck path. St000058The order of a permutation. St000086The number of subgraphs. St000089The absolute variation of a composition. St000468The Hosoya index of a graph. St000797The stat`` of a permutation. St000883The number of longest increasing subsequences of a permutation. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001313The number of Dyck paths above the lattice path given by a binary word. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St000728The dimension of a set partition. St000216The absolute length of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000454The largest eigenvalue of a graph if it is integral. St000472The sum of the ascent bottoms of a permutation. St000653The last descent of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. 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). St001077The prefix exchange distance of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001644The dimension of a graph. St000100The number of linear extensions of a poset. St000420The number of Dyck paths that are weakly above a Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001330The hat guessing number of a graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000495The number of inversions of distance at most 2 of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000210Minimum over maximum difference of elements in cycles. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000239The number of small weak excedances. St000297The number of leading ones in a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000488The number of cycles of a permutation of length at most 2. St000631The number of distinct palindromic decompositions of a binary word. St000682The Grundy value of Welter's game on a binary word. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001480The number of simple summands of the module J^2/J^3. St001596The number of two-by-two squares inside a skew partition. St001821The sorting index of a signed permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St001956The comajor index for set-valued two-row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000120The number of left tunnels of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000006The dinv of a Dyck path. St000766The number of inversions of an integer composition. 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. St000691The number of changes of a binary word. St001267The length of the Lyndon factorization of the binary word. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000668The least common multiple of the parts of the partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000567The sum of the products of all pairs of parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000082The number of elements smaller than a binary tree in Tamari order. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000247The number of singleton blocks of a set partition. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000444The length of the maximal rise of a Dyck path. St000477The weight of a partition according to Alladi. St000625The sum of the minimal distances to a greater element. St000656The number of cuts of a poset. St000673The number of non-fixed points of a permutation. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000680The Grundy value for Hackendot on posets. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000792The Grundy value for the game of ruler on a binary word. St000830The total displacement of a permutation. St000925The number of topologically connected components of a set partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000956The maximal displacement of a permutation. St000990The first ascent of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001346The number of parking functions that give the same permutation. St001394The genus of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000141The maximum drop size of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St001096The size of the overlap set of a permutation. St000422The energy of a graph, if it is integral. 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 $eA$. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000014The number of parking functions supported by a Dyck path. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000063The number of linear extensions of a certain poset defined for an integer partition. St000091The descent variation of a composition. St000144The pyramid weight of the Dyck path. St000173The segment statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000317The cycle descent number of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000491The number of inversions of a set partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000626The minimal period of a binary word. St000650The number of 3-rises of a permutation. St000759The smallest missing part in an integer partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001400The total number of Littlewood-Richardson tableaux of given shape. St001403The number of vertical separators in a permutation. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001781The interlacing number of a set partition. St001814The number of partitions interlacing the given partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001843The Z-index of a set partition. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000314The number of left-to-right-maxima of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001461The number of topologically connected components of the chord diagram of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000456The monochromatic index of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001176The size of a partition minus its first part. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. 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. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000834The number of right outer peaks of a permutation. St001645The pebbling number of a connected graph. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000219The number of occurrences of the pattern 231 in a permutation. St000365The number of double ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. 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. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000707The product of the factorials of the parts. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. 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 $eA$. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001948The number of augmented double ascents of a permutation. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000779The tier of a permutation. St001469The holeyness of a permutation. St001487The number of inner corners of a skew partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001557The number of inversions of the second entry of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. 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. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000137The Grundy value of an integer partition. St000177The number of free tiles in the pattern. St000178Number of free entries. St000284The Plancherel distribution on integer partitions. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000418The number of Dyck paths that are weakly below a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000438The position of the last up step in a Dyck path. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000675The number of centered multitunnels of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. 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. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St000981The length of the longest zigzag subpath. St000984The number of boxes below precisely one peak. St000997The even-odd crank of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001118The acyclic chromatic index of a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St001383The BG-rank of an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. 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. St001531Number of partial orders contained in the poset determined by the Dyck path. 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. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001811The Castelnuovo-Mumford regularity of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000717The number of ordinal summands of a poset.
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