Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001683
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,1,2] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,4,1,3] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => 0
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000204
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [.,.]
 => 0
{{1,2}}
 => [2,1] => [2,1] => [[.,.],.]
 => 0
{{1},{2}}
 => [1,2] => [1,2] => [.,[.,.]]
 => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [[.,.],[.,.]]
 => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 0
Description
The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000356
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,1,2] => [3,4,2,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => [2,4,3,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => [2,4,1,3] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000358
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => [2,3,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => [3,1,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,1,3] => [3,2,4,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [3,2,4,1] => [4,2,1,3] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,1,3] => [3,1,4,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001882
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [2,1,3] => [2,1,3] => 0
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => [2,3,1] => 1
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,1,3] => [4,2,1,3] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [2,1,3,4] => [2,1,3,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,3,1,4] => [2,3,1,4] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,4,1,3] => [2,4,1,3] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St000886
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000886: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0 + 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [1,3,2] => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1 = 0 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,3,1,2] => [1,4,3,2] => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => [4,1,3,2] => 2 = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => [1,3,4,2] => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 3 = 2 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1 = 0 + 1
Description
The number of permutations with the same antidiagonal sums. The X-ray of a permutation $\pi$ is the vector of the sums of the antidiagonals of the permutation matrix of $\pi$, read from left to right. For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right),$$ so its X-ray is $(0, 1, 1, 1, 0, 0, 0, 2, 0)$. This statistic records the number of permutations having the same X-ray as the given permutation. In [1] this is called the degeneracy of the X-ray of the permutation. By [prop.1, 1], the number of different X-rays of permutations of size $n$ equals the number of nondecreasing differences of permutations of size $n$, [2].
Matching statistic: St001857
(load all 6 compositions to match this statistic)
Mapping
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => {{1}}
 => [1] => [1] => 0
{{1,2}}
 => {{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => {{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => ? = 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => ? = 0
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ? = 0
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => ? = 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ? = 0
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => ? = 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => ? = 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ? = 0
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => ? = 0
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => ? = 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => ? = 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ? = 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => ? = 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => ? = 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ? = 0
{{1,5},{2,4},{3}}
 => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [1,4,5,2,3] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St001876
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => ([],1)
 => ([],1)
 => ? = 0
{{1,2}}
 => [2,1] => ([],2)
 => ([],1)
 => ? = 0
{{1},{2}}
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0
{{1,2,3}}
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0
{{1,2},{3}}
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? = 0
{{1,3},{2}}
 => [3,2,1] => ([],3)
 => ([],1)
 => ? = 0
{{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? = 1
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
{{1,2,4},{3}}
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? = 0
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? = 0
{{1,3,4},{2}}
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? = 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0
{{1,4},{2,3}}
 => [4,3,2,1] => ([],4)
 => ([],1)
 => ? = 0
{{1},{2,3,4}}
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? = 2
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([],5)
 => ([],1)
 => ? = 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => ([],1)
 => ([],1)
 => ? = 0
{{1,2}}
 => [2,1] => ([],2)
 => ([],1)
 => ? = 0
{{1},{2}}
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0
{{1,2,3}}
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0
{{1,2},{3}}
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? = 0
{{1,3},{2}}
 => [3,2,1] => ([],3)
 => ([],1)
 => ? = 0
{{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? = 1
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
{{1,2,4},{3}}
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? = 0
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? = 0
{{1,3,4},{2}}
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? = 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0
{{1,4},{2,3}}
 => [4,3,2,1] => ([],4)
 => ([],1)
 => ? = 0
{{1},{2,3,4}}
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? = 2
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([],5)
 => ([],1)
 => ? = 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001878
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => ([],1)
 => ([],1)
 => ? = 0 + 1
{{1,2}}
 => [2,1] => ([],2)
 => ([],1)
 => ? = 0 + 1
{{1},{2}}
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0 + 1
{{1,2,3}}
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 1
{{1,2},{3}}
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? = 0 + 1
{{1,3},{2}}
 => [3,2,1] => ([],3)
 => ([],1)
 => ? = 0 + 1
{{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => ([],4)
 => ([],1)
 => ? = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? = 2 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([],5)
 => ([],1)
 => ? = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000102The charge of a semistandard tableau. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.