- FindStat

Your data matches 246 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001052
(load all 4 compositions to match this statistic)

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001052: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1 = 3 - 2

Description

The length of the exterior of a permutation. The '''exterior''' of a permutation is the longest proper prefix that is also a suffix, when viewed as a pattern. In other words, the length of the exterior of a permutation

$\sigma$ of length

$n$ is the largest

$i < n$ such that the first

$i$ entries of

$\sigma$ are in the same relative order as the last

$i$ entries of

$\sigma$ .

Matching statistic: St001096
(load all 4 compositions to match this statistic)

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St001096: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1 = 3 - 2

Description

The size of the overlap set of a permutation. For a permutation

$\pi\in\mathfrak S_n$ this is the number of indices

$i < n$ such that the standardisation of

$\pi_1\dots\pi_{n-i}$ equals the standardisation of

$\pi_{i+1}\dots\pi_n$ . In particular, for

$n > 1$ , the statistic is at least one, because the standardisations of

$\pi_1$ and

$\pi_n$ are both

$1$ . For example, for

$\pi=2143$ , the standardisations of

$21$ and

$43$ are equal, and so are the standardisations of

$2$ and

$3$ . Thus, the statistic on

$\pi$ is

$2$ .

Matching statistic: St000264
(load all 2 compositions to match this statistic)

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St000759

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [3,1,1]
 => 2 = 3 - 1
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [3,1,1]
 => 2 = 3 - 1
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [3,1,1]
 => 2 = 3 - 1
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [3,1,1]
 => 2 = 3 - 1
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,1,1,1,1]
 => 3 = 4 - 1
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,1,1,1,1]
 => 3 = 4 - 1
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,1,1,1,1]
 => 3 = 4 - 1
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => [3,1]
 => 2 = 3 - 1
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,1,1,1]
 => 3 = 4 - 1
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => [3,1,1]
 => 2 = 3 - 1
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,1,1]
 => 2 = 3 - 1

Description

The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].

Matching statistic: St001184

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1

Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Matching statistic: St000326

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => 1100 => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => 1100 => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => 1100 => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => 1100 => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 01000 => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 01000 => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => 01000 => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 110 => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 0100 => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => 1100 => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 1100 => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 1100 => 1 = 3 - 2

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000382

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,4] => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,4] => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,4] => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,3] => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2

Description

The first part of an integer composition.

Matching statistic: St000657

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,4] => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,4] => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [5,6,1,2,3,4] => [2,4] => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => [1,1,2] => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,3] => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 1 = 3 - 2

Description

The smallest part of an integer composition.

Matching statistic: St000667

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [2]
 => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,2]
 => [2]
 => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2

Description

The greatest common divisor of the parts of the partition.

Matching statistic: St000993

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,5],[2],[3]]
 => [[1,2],[3],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,5],[2],[4]]
 => [[1,2],[4],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,5],[3],[4]]
 => [[1,3],[4],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2,5],[3],[4]]
 => [[2,3],[4],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,1,4],[2],[3]]
 => [[1,1,2],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,2,4],[2],[3]]
 => [[1,2,2],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,3,4],[2],[3]]
 => [[1,2,3],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,4,4],[2],[3]]
 => [[1,2,4],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,1],[2],[3],[4]]
 => [[1,1,2],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,2],[2],[3],[4]]
 => [[1,2,2],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,3],[2],[3],[4]]
 => [[1,2,3],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,4],[2],[3],[4]]
 => [[1,2,4],[3],[4]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,1,3,3],[2,2]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [2,2,1,1]
 => 2 = 4 - 2
[[1,1,3],[2,2],[3]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [2,2,1,1]
 => 2 = 4 - 2
[[1,1],[2,2],[3,3]]
 => [[1,1,2,2],[3,3]]
 => [4,2]
 => [2,2,1,1]
 => 2 = 4 - 2
[[1,6],[2],[3]]
 => [[1,2],[3],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,6],[2],[4]]
 => [[1,2],[4],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,6],[2],[5]]
 => [[1,2],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,6],[3],[4]]
 => [[1,3],[4],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,6],[3],[5]]
 => [[1,3],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,6],[4],[5]]
 => [[1,4],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2,6],[3],[4]]
 => [[2,3],[4],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2,6],[3],[5]]
 => [[2,3],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2,6],[4],[5]]
 => [[2,4],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[3,6],[4],[5]]
 => [[3,4],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [2,1,1]
 => [3,1]
 => 1 = 3 - 2
[[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => [3,2]
 => [2,2,1]
 => 2 = 4 - 2
[[1,1,5],[2],[3]]
 => [[1,1,2],[3],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,1,5],[2],[4]]
 => [[1,1,2],[4],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,1,5],[3],[4]]
 => [[1,1,3],[4],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,2,5],[2],[3]]
 => [[1,2,2],[3],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,2,5],[2],[4]]
 => [[1,2,2],[4],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,3,5],[2],[3]]
 => [[1,2,3],[3],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,2,5],[3],[4]]
 => [[1,2,3],[4],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [3,1,1]
 => [3,1,1]
 => 1 = 3 - 2

Description

The multiplicity of the largest part of an integer partition.

The following 236 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001571The Cartan determinant of the integer partition. St001884The number of borders of a binary word. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001657The number of twos in an integer partition. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001665The number of pure excedances of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001330The hat guessing number of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St001856The number of edges in the reduced word graph of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001948The number of augmented double ascents of a permutation. St001684The reduced word complexity of a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000068The number of minimal elements in a poset. St001429The number of negative entries in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001861The number of Bruhat lower covers of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001864The number of excedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001894The depth of a signed permutation. St000735The last entry on the main diagonal of a standard tableau. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001556The number of inversions of the third entry of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000633The size of the automorphism group of a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001399The distinguishing number of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000850The number of 1/2-balanced pairs in a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000454The largest eigenvalue of a graph if it is integral. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000422The energy of a graph, if it is integral. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001863The number of weak excedances of a signed permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001817The number of flag weak exceedances of a signed permutation. St001071The beta invariant of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001354The number of series nodes in the modular decomposition of a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001867The number of alignments of type EN of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000135The number of lucky cars of the parking function. St001822The number of alignments of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000540The sum of the entries of a parking function minus its length. St000943The number of spots the most unlucky car had to go further in a parking function. St000165The sum of the entries of a parking function. St001852The size of the conjugacy class of the signed permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000080The rank of the poset. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000942The number of critical left to right maxima of the parking functions. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001209The pmaj statistic of a parking function. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000298The order dimension or Dushnik-Miller dimension of a poset. St000528The height of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000744The length of the path to the largest entry in a standard Young tableau. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000907The number of maximal antichains of minimal length in a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001433The flag major index of a signed permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001768The number of reduced words of a signed permutation. St001819The flag Denert index of a signed permutation. St001821The sorting index of a signed permutation. St001893The flag descent of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000044The number of vertices of the unicellular map given by a perfect matching. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001268The size of the largest ordinal summand in the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001557The number of inversions of the second entry of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001718The number of non-empty open intervals in a poset. St001770The number of facets of a certain subword complex associated with the signed permutation. St001854The size of the left Kazhdan-Lusztig cell, St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St000017The number of inversions of a standard tableau. St000072The number of circled entries. St000073The number of boxed entries. St000077The number of boxed and circled entries. St000189The number of elements in the poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001343The dimension of the reduced incidence algebra of a poset. St001371The length of the longest Yamanouchi prefix of a binary word. St001434The number of negative sum pairs of a signed permutation. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001717The largest size of an interval in a poset. St001730The number of times the path corresponding to a binary word crosses the base line. St001851The number of Hecke atoms of a signed permutation. St001902The number of potential covers of a poset. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001721The degree of a binary word. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000656The number of cuts of a poset. St001848The atomic length of a signed permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001782The order of rowmotion on the set of order ideals of a poset. St001858The number of covering elements of a signed permutation in absolute order. St001865The number of alignments of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000070The number of antichains in a poset. St001779The order of promotion on the set of linear extensions of a poset. St000016The number of attacking pairs of a standard tableau. St000641The number of non-empty boolean intervals in a poset. St000639The number of relations in a poset. St000958The number of Bruhat factorizations of a permutation. St001853The size of the two-sided Kazhdan-Lusztig cell, St001664The number of non-isomorphic subposets of a poset. St001885The number of binary words with the same proper border set. St001709The number of homomorphisms to the three element chain of a poset. St001815The number of order preserving surjections from a poset to a total order. St001813The product of the sizes of the principal order filters in a poset. St001644The dimension of a graph. St000081The number of edges of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St000822The Hadwiger number of the graph. St001642The Prague dimension of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001734The lettericity of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001964The interval resolution global dimension of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.