Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000496
(load all 10 compositions to match this statistic)
Mapping
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => 0
{{1,2}}
 => 0
{{1},{2}}
 => 0
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 0
{{1,3},{2}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 2
{{1,3},{2,4}}
 => 0
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1},{2},{3},{4},{5}}
 => 0
Description
The rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000057
(load all 2 compositions to match this statistic)
Mapping
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => {{1}}
 => [[1]]
 => 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => [[1,2]]
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => [[1],[2]]
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => [[1,2,3]]
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => {{1},{2,3}}
 => [[1,3],[2]]
 => 0
{{1,3},{2}}
 => {{1},{2,3}}
 => {{1,2},{3}}
 => [[1,2],[3]]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => [[1,3],[2]]
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => [[1],[2],[3]]
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => [[1,2,3,4]]
 => 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 0
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 0
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 0
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 2
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 0
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 1
{{1,4},{2,3}}
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 1
{{1},{2,3,4}}
 => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 0
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 0
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 2
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [[1],[2],[3],[4],[5]]
 => 0
Description
The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000358
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ 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] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [3,1,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [4,3,1,2] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{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] => [2,4,3,1] => [4,1,3,2] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [4,2,1,3] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [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: St000359
(load all 2 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => [3,2,1] => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 0
{{1,3},{2}}
 => {{1,3},{2}}
 => [3,2,1] => [2,3,1] => 1
{{1},{2,3}}
 => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{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] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => 0
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 1
{{1,4},{2,3}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 2
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 1
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St001683
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001683: 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] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [1,3,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => [1,3,4,2] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [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: St001771
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
St001771: 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] => [2,1] => [-2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [-3,-2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [-2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,-3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [-3,1,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => [3,-4,-2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => 0
{{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] => [3,2,4,1] => [2,-4,-3,1] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001882
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ 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] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{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] => [2,4,3,1] => [2,4,3,1] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [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: St000589
(load all 4 compositions to match this statistic)
Mapping
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1,3},{2}}
 => {{1},{2,3}}
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 0
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => 2
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1,4},{2,3}}
 => {{1,3,4},{2}}
 => 1
{{1},{2,3,4}}
 => {{1,3},{2,4}}
 => 0
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 2
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000607
(load all 4 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000607: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => {{1}}
 => ? = 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 1
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 0
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1},{2,3,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1,4},{2,3}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 2
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000936
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%
Values
{{1}}
 => [1] => [1]
 => []
 => ? = 0
{{1,2}}
 => [2,1] => [2]
 => []
 => ? = 0
{{1},{2}}
 => [1,2] => [1,1]
 => [1]
 => ? = 0
{{1,2,3}}
 => [2,3,1] => [2,1]
 => [1]
 => ? = 0
{{1,2},{3}}
 => [2,1,3] => [2,1]
 => [1]
 => ? = 0
{{1,3},{2}}
 => [3,2,1] => [3]
 => []
 => ? = 1
{{1},{2,3}}
 => [1,3,2] => [2,1]
 => [1]
 => ? = 0
{{1},{2},{3}}
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1]
 => [1]
 => ? = 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,2]
 => [2]
 => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,1]
 => [1]
 => ? = 2
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,1]
 => [1]
 => ? = 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4]
 => []
 => ? = 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1]
 => [1]
 => ? = 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
Description
The number of even values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$. It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St001857The number of edges in the reduced word graph of a signed permutation. 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.