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Your data matches 243 different statistics following compositions of up to 3 maps.
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Matching statistic: St001122
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(load all 9 compositions to match this statistic)
St001122: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 0 = 1 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 0 = 1 - 1
[3,1]
=> 0 = 1 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 0 = 1 - 1
[1,1,1,1]
=> 0 = 1 - 1
[5]
=> 0 = 1 - 1
[4,1]
=> 0 = 1 - 1
[3,2]
=> 0 = 1 - 1
[3,1,1]
=> 1 = 2 - 1
[2,2,1]
=> 0 = 1 - 1
[2,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> 0 = 1 - 1
[5,1]
=> 0 = 1 - 1
[4,2]
=> 0 = 1 - 1
[4,1,1]
=> 0 = 1 - 1
[3,3]
=> 0 = 1 - 1
[3,2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> 0 = 1 - 1
[2,2,1,1]
=> 0 = 1 - 1
[2,1,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> 0 = 1 - 1
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St000633
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00185: Skew partitions —cell poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 1
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 2
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The size of the automorphism group of a poset.
A poset automorphism is a permutation of the elements of the poset preserving the order relation.
Matching statistic: St001399
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00185: Skew partitions —cell poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 1
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 2
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The distinguishing number of a poset.
This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring.
See also [[St000469]], which is the same concept for graphs.
Matching statistic: St001263
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001263: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001263: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => 0 = 1 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1] => 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => 0 = 1 - 1
Description
The index of the maximal parabolic seaweed algebra associated with the composition.
Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows:
* place $n$ dots on a horizontal line
* subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line
* subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$
* within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line
By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is
$$
\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1,
$$
where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander.
This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.
Matching statistic: St000899
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => [2,1] => 1
[1,1]
=> 110 => 011 => [1,2] => 1
[3]
=> 1000 => 0001 => [3,1] => 1
[2,1]
=> 1010 => 0011 => [2,2] => 2
[1,1,1]
=> 1110 => 0111 => [1,3] => 1
[4]
=> 10000 => 00001 => [4,1] => 1
[3,1]
=> 10010 => 00011 => [3,2] => 1
[2,2]
=> 1100 => 0011 => [2,2] => 2
[2,1,1]
=> 10110 => 00111 => [2,3] => 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 1
[5]
=> 100000 => 000001 => [5,1] => 1
[4,1]
=> 100010 => 000011 => [4,2] => 1
[3,2]
=> 10100 => 00011 => [3,2] => 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 2
[2,2,1]
=> 11010 => 00111 => [2,3] => 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 1
[6]
=> 1000000 => 0000001 => [6,1] => 1
[5,1]
=> 1000010 => 0000011 => [5,2] => 1
[4,2]
=> 100100 => 000011 => [4,2] => 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 1
[3,3]
=> 11000 => 00011 => [3,2] => 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 2
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 1
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000902
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => [2,1] => 1
[1,1]
=> 110 => 011 => [1,2] => 1
[3]
=> 1000 => 0001 => [3,1] => 1
[2,1]
=> 1010 => 0011 => [2,2] => 2
[1,1,1]
=> 1110 => 0111 => [1,3] => 1
[4]
=> 10000 => 00001 => [4,1] => 1
[3,1]
=> 10010 => 00011 => [3,2] => 1
[2,2]
=> 1100 => 0011 => [2,2] => 2
[2,1,1]
=> 10110 => 00111 => [2,3] => 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 1
[5]
=> 100000 => 000001 => [5,1] => 1
[4,1]
=> 100010 => 000011 => [4,2] => 1
[3,2]
=> 10100 => 00011 => [3,2] => 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 2
[2,2,1]
=> 11010 => 00111 => [2,3] => 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 1
[6]
=> 1000000 => 0000001 => [6,1] => 1
[5,1]
=> 1000010 => 0000011 => [5,2] => 1
[4,2]
=> 100100 => 000011 => [4,2] => 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 1
[3,3]
=> 11000 => 00011 => [3,2] => 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 2
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 1
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000904
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => [2,1] => 1
[1,1]
=> 110 => 011 => [1,2] => 1
[3]
=> 1000 => 0001 => [3,1] => 1
[2,1]
=> 1010 => 0011 => [2,2] => 2
[1,1,1]
=> 1110 => 0111 => [1,3] => 1
[4]
=> 10000 => 00001 => [4,1] => 1
[3,1]
=> 10010 => 00011 => [3,2] => 1
[2,2]
=> 1100 => 0011 => [2,2] => 2
[2,1,1]
=> 10110 => 00111 => [2,3] => 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 1
[5]
=> 100000 => 000001 => [5,1] => 1
[4,1]
=> 100010 => 000011 => [4,2] => 1
[3,2]
=> 10100 => 00011 => [3,2] => 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 2
[2,2,1]
=> 11010 => 00111 => [2,3] => 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 1
[6]
=> 1000000 => 0000001 => [6,1] => 1
[5,1]
=> 1000010 => 0000011 => [5,2] => 1
[4,2]
=> 100100 => 000011 => [4,2] => 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 1
[3,3]
=> 11000 => 00011 => [3,2] => 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 2
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 1
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001850
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001850: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001850: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,7,5] => 0 = 1 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,2,5,3,6,4] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,2,3,4,7,5,6] => 0 = 1 - 1
Description
The number of Hecke atoms of a permutation.
For a permutation $z\in\mathfrak S_n$, this is the cardinality of the set
$$
\{ w\in\mathfrak S_n | w^{-1} \star w = z\},
$$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
Matching statistic: St001940
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 0 = 1 - 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 0 = 1 - 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 0 = 1 - 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 1 = 2 - 1
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 0 = 1 - 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 0 = 1 - 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 0 = 1 - 1
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 1 = 2 - 1
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 0 = 1 - 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0 = 1 - 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 0 = 1 - 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 0 = 1 - 1
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 0 = 1 - 1
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 1 = 2 - 1
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 0 = 1 - 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 0 = 1 - 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 0 = 1 - 1
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 0 = 1 - 1
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 0 = 1 - 1
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 0 = 1 - 1
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 1 = 2 - 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 0 = 1 - 1
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 0 = 1 - 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St000791
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[1,1]
=> [2]
=> []
=> []
=> ? = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1]
=> [3]
=> []
=> []
=> ? = 2 - 1
[1,1,1]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,2]
=> [4]
=> []
=> []
=> ? = 2 - 1
[2,1,1]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,2]
=> [5]
=> []
=> []
=> ? = 1 - 1
[3,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,3]
=> [6]
=> []
=> []
=> ? = 1 - 1
[3,2,1]
=> [5,1]
=> [1]
=> [1,0]
=> ? = 2 - 1
[3,1,1,1]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,2,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
The following 233 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000264The girth of a graph, which is not a tree. St000741The Colin de Verdière graph invariant. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001851The number of Hecke atoms of a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000735The last entry on the main diagonal of a standard tableau. 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 $eA$. St000369The dinv deficit of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000075The orbit size of a standard tableau under promotion. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000365The number of double ascents of a permutation. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. 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 $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001568The smallest positive integer that does not appear twice in the partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001488The number of corners of a skew partition. St001623The number of doubly irreducible elements of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000068The number of minimal elements in a poset. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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 $K[x]/(x^n)$. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000234The number of global ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000486The number of cycles of length at least 3 of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000729The minimal arc length of a set partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001712The number of natural descents of a standard Young tableau. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000222The number of alignments in the permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000941The number of characters of the symmetric group whose value on the partition is even. St000991The number of right-to-left minima of a permutation. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001535The number of cyclic alignments of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001841The number of inversions of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001911A descent variant minus the number of inversions. St001928The number of non-overlapping descents in a permutation. St000080The rank of the poset. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000570The Edelman-Greene number of a permutation. St000572The dimension exponent of a set partition. St000824The sum of the number of descents and the number of recoils of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001424The number of distinct squares in a binary word. St001760The number of prefix or suffix reversals needed to sort a permutation. St000519The largest length of a factor maximising the subword complexity. St000677The standardized bi-alternating inversion number of a permutation. St000906The length of the shortest maximal chain in a poset. 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. St001375The pancake length of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001705The number of occurrences of the pattern 2413 in a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000304The load of a permutation. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001077The prefix exchange distance of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001718The number of non-empty open intervals in a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001848The atomic length of a signed permutation. St001377The major index minus the number of inversions of a permutation. St000728The dimension of a set partition. St000008The major index of the composition. St000154The sum of the descent bottoms of a permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000305The inverse major index of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001671Haglund's hag of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000391The sum of the positions of the ones in a binary word. St001684The reduced word complexity of a permutation. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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