searching the database
Your data matches 126 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000174
St000174: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0
[[2,2]]
=> 1
[[1],[2]]
=> 0
[[1,3]]
=> 0
[[2,3]]
=> 1
[[3,3]]
=> 1
[[1],[3]]
=> 1
[[2],[3]]
=> 2
[[1,1,2]]
=> 0
[[1,2,2]]
=> 0
[[2,2,2]]
=> 1
[[1,1],[2]]
=> 0
[[1,2],[2]]
=> 1
[[1,4]]
=> 0
[[2,4]]
=> 1
[[3,4]]
=> 1
[[4,4]]
=> 1
[[1],[4]]
=> 1
[[2],[4]]
=> 2
[[3],[4]]
=> 2
[[1,1,3]]
=> 0
[[1,2,3]]
=> 0
[[1,3,3]]
=> 0
[[2,2,3]]
=> 1
[[2,3,3]]
=> 1
[[3,3,3]]
=> 1
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 1
[[1,3],[2]]
=> 1
[[1,3],[3]]
=> 2
[[2,2],[3]]
=> 2
[[2,3],[3]]
=> 3
[[1],[2],[3]]
=> 0
[[1,1,1,2]]
=> 0
[[1,1,2,2]]
=> 0
[[1,2,2,2]]
=> 0
[[2,2,2,2]]
=> 1
[[1,1,1],[2]]
=> 0
[[1,1,2],[2]]
=> 0
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 0
[[1,1,4]]
=> 0
[[1,2,4]]
=> 0
[[1,3,4]]
=> 0
[[1,4,4]]
=> 0
[[2,2,4]]
=> 1
[[2,3,4]]
=> 1
[[2,4,4]]
=> 1
[[3,3,4]]
=> 1
[[3,4,4]]
=> 1
Description
The flush statistic of a semistandard tableau.
Let $T$ be a tableaux with $r$ rows such that each row is longer than the row beneath it by at least one box. Let $1 \leq i < k \leq r+1$ and suppose $l$ is the smallest integer greater than $k$ such that there exists an $l$-segment in the $(i+1)$-st row of $T$. A $k$-segment in the $i$-th row of $T$ is called '''flush''' if the leftmost box in the $k$-segment and the leftmost box of the $l$-segment are in the same column of $T$. If, however, no such $l$ exists, then this $k$-segment is said to be flush if the number of boxes in the $k$-segment is equal to difference of the number of boxes between the $i$-th row and $(i+1)$-st row. The flush statistic is given by the number of $k$-segments in $T$.
Matching statistic: St000510
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 50%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 50%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000681
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1}
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000260
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 38%●distinct values known / distinct values provided: 29%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 38%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2}
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2}
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2}
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1}
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,1}
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2,2}
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2,2}
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2,2}
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1,2,2}
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,3}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0}
[[1,1,1],[2]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0}
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,1],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3],[4]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,3}
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,1,1],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,2],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,2],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,2],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,2,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,3,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[3,3,3],[4]]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,4]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001199
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 43%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0}
[[2,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0}
[[1],[2]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2}
[[2,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2}
[[3,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2}
[[1],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[2],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1}
[[1,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2,2}
[[2,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2,2}
[[3,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2,2}
[[4,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,1,2,2}
[[1],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[2],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[3],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3}
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0}
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0}
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0}
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0}
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0}
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0}
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,1,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[2,2,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[2,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[3,3,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[3,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[4,4,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1,1],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,4],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1,3],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,4],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1,4],[4]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[2,2],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[2,3],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[2,4],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[2,4],[4]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[3,3],[4]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[3,4],[4]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3}
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,1,2,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,1,3,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,2,2,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,2,3,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,3,3,3]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1,2],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2,2],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,1,1],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1,2],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2,2],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[2,2,2],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[2,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[2,3,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[3,3,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,1],[2,4]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000567
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[1,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1,1,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000620
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[1,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1,1,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000668
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[1,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1,1,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[1,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1,1,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
Matching statistic: St000708
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1}
[[1],[2]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1}
[[1,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2,3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2}
[[1],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[2],[3]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2}
[[1,1,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1}
[[1,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3,4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[[1,1,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2,3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3,3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,3}
[[1,1,1,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2,2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,2],[2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
[[1,1,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,2,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,3,4]]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[2,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,3,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[3,4,4]]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[[1,2,3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,4],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,4],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,4],[2]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2,4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,3],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The product of the parts of an integer partition.
The following 116 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. 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. St000137The Grundy value of an integer partition. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. St001609The number of coloured trees 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. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001568The smallest positive integer that does not appear twice in the partition. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St000454The largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. 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. 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. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001123The multiplicity of the dual of the standard 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!