- FindStat

Your data matches 257 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000259
(load all 8 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => 1
[[1],[3]]
 => [2,1] => ([(0,1)],2)
 => 1
[[2],[3]]
 => [2,1] => ([(0,1)],2)
 => 1
[[1],[4]]
 => [2,1] => ([(0,1)],2)
 => 1
[[2],[4]]
 => [2,1] => ([(0,1)],2)
 => 1
[[3],[4]]
 => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[5]]
 => [2,1] => ([(0,1)],2)
 => 1
[[2],[5]]
 => [2,1] => ([(0,1)],2)
 => 1
[[3],[5]]
 => [2,1] => ([(0,1)],2)
 => 1
[[4],[5]]
 => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[3],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[3],[4]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1],[6]]
 => [2,1] => ([(0,1)],2)
 => 1
[[2],[6]]
 => [2,1] => ([(0,1)],2)
 => 1
[[3],[6]]
 => [2,1] => ([(0,1)],2)
 => 1
[[4],[6]]
 => [2,1] => ([(0,1)],2)
 => 1
[[5],[6]]
 => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[5]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[3],[5]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[4],[5]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[3],[5]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[4],[5]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[3],[4],[5]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2
[[1],[7]]
 => [2,1] => ([(0,1)],2)
 => 1
[[2],[7]]
 => [2,1] => ([(0,1)],2)
 => 1
[[3],[7]]
 => [2,1] => ([(0,1)],2)
 => 1
[[4],[7]]
 => [2,1] => ([(0,1)],2)
 => 1
[[5],[7]]
 => [2,1] => ([(0,1)],2)
 => 1
[[6],[7]]
 => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[3],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[4],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[5],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[3],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[4],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[5],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[3],[4],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[3],[5],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[4],[5],[6]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St000374
(load all 13 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [2,1] => 1
[[1],[3]]
 => [2,1] => [2,1] => 1
[[2],[3]]
 => [2,1] => [2,1] => 1
[[1],[4]]
 => [2,1] => [2,1] => 1
[[2],[4]]
 => [2,1] => [2,1] => 1
[[3],[4]]
 => [2,1] => [2,1] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[[1],[5]]
 => [2,1] => [2,1] => 1
[[2],[5]]
 => [2,1] => [2,1] => 1
[[3],[5]]
 => [2,1] => [2,1] => 1
[[4],[5]]
 => [2,1] => [2,1] => 1
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => 1
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => 1
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [4,2,1,3] => 2
[[1],[6]]
 => [2,1] => [2,1] => 1
[[2],[6]]
 => [2,1] => [2,1] => 1
[[3],[6]]
 => [2,1] => [2,1] => 1
[[4],[6]]
 => [2,1] => [2,1] => 1
[[5],[6]]
 => [2,1] => [2,1] => 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [4,2,1,3] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [4,2,1,3] => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [4,2,1,3] => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [4,2,1,3] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [4,2,5,1,3] => 2
[[1],[7]]
 => [2,1] => [2,1] => 1
[[2],[7]]
 => [2,1] => [2,1] => 1
[[3],[7]]
 => [2,1] => [2,1] => 1
[[4],[7]]
 => [2,1] => [2,1] => 1
[[5],[7]]
 => [2,1] => [2,1] => 1
[[6],[7]]
 => [2,1] => [2,1] => 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [4,2,1,3] => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [4,2,1,3] => 2

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000451
(load all 13 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[3]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[2],[3]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[4]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[2],[4]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[3],[4]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[5]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[2],[5]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[3],[5]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[4],[5]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[[1],[6]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[2],[6]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[3],[6]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[4],[6]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[5],[6]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[[1,3],[3],[4]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[[2,3],[3],[4]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [4,2,5,1,3] => 3 = 2 + 1
[[1],[7]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[2],[7]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[3],[7]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[4],[7]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[5],[7]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[6],[7]]
 => [2,1] => [2,1] => 2 = 1 + 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 1 + 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[[1,3],[2],[5]]
 => [4,2,1,3] => [4,2,1,3] => 3 = 2 + 1

Description

The length of the longest pattern of the form k 1 2...(k-1).

Matching statistic: St000007
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [1] => [1] => 1
[[1],[3]]
 => [2,1] => [1] => [1] => 1
[[2],[3]]
 => [2,1] => [1] => [1] => 1
[[1],[4]]
 => [2,1] => [1] => [1] => 1
[[2],[4]]
 => [2,1] => [1] => [1] => 1
[[3],[4]]
 => [2,1] => [1] => [1] => 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[5]]
 => [2,1] => [1] => [1] => 1
[[2],[5]]
 => [2,1] => [1] => [1] => 1
[[3],[5]]
 => [2,1] => [1] => [1] => 1
[[4],[5]]
 => [2,1] => [1] => [1] => 1
[[1],[2],[4]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[3],[4]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[2],[3],[4]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2
[[1],[6]]
 => [2,1] => [1] => [1] => 1
[[2],[6]]
 => [2,1] => [1] => [1] => 1
[[3],[6]]
 => [2,1] => [1] => [1] => 1
[[4],[6]]
 => [2,1] => [1] => [1] => 1
[[5],[6]]
 => [2,1] => [1] => [1] => 1
[[1],[2],[5]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[3],[5]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[4],[5]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[2],[3],[5]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[2],[4],[5]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[3],[4],[5]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1] => [2,1,3] => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[[1],[7]]
 => [2,1] => [1] => [1] => 1
[[2],[7]]
 => [2,1] => [1] => [1] => 1
[[3],[7]]
 => [2,1] => [1] => [1] => 1
[[4],[7]]
 => [2,1] => [1] => [1] => 1
[[5],[7]]
 => [2,1] => [1] => [1] => 1
[[6],[7]]
 => [2,1] => [1] => [1] => 1
[[1],[2],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[3],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[4],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1],[5],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[2],[3],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[2],[4],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[2],[5],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[3],[4],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[3],[5],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[4],[5],[6]]
 => [3,2,1] => [2,1] => [1,2] => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [2,1,3] => [1,3,2] => 2

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000028
(load all 3 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [2,1] => [2,1] => 1
[[1],[3]]
 => [2,1] => [2,1] => [2,1] => 1
[[2],[3]]
 => [2,1] => [2,1] => [2,1] => 1
[[1],[4]]
 => [2,1] => [2,1] => [2,1] => 1
[[2],[4]]
 => [2,1] => [2,1] => [2,1] => 1
[[3],[4]]
 => [2,1] => [2,1] => [2,1] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[5]]
 => [2,1] => [2,1] => [2,1] => 1
[[2],[5]]
 => [2,1] => [2,1] => [2,1] => 1
[[3],[5]]
 => [2,1] => [2,1] => [2,1] => 1
[[4],[5]]
 => [2,1] => [2,1] => [2,1] => 1
[[1],[2],[4]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[3],[4]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[2],[3],[4]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1],[6]]
 => [2,1] => [2,1] => [2,1] => 1
[[2],[6]]
 => [2,1] => [2,1] => [2,1] => 1
[[3],[6]]
 => [2,1] => [2,1] => [2,1] => 1
[[4],[6]]
 => [2,1] => [2,1] => [2,1] => 1
[[5],[6]]
 => [2,1] => [2,1] => [2,1] => 1
[[1],[2],[5]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[3],[5]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[4],[5]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[2],[3],[5]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[2],[4],[5]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[3],[4],[5]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,1,2,3] => [4,1,3,2] => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [3,4,5,2,1] => [3,5,4,2,1] => 2
[[1],[7]]
 => [2,1] => [2,1] => [2,1] => 1
[[2],[7]]
 => [2,1] => [2,1] => [2,1] => 1
[[3],[7]]
 => [2,1] => [2,1] => [2,1] => 1
[[4],[7]]
 => [2,1] => [2,1] => [2,1] => 1
[[5],[7]]
 => [2,1] => [2,1] => [2,1] => 1
[[6],[7]]
 => [2,1] => [2,1] => [2,1] => 1
[[1],[2],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[3],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[4],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1],[5],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[2],[3],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[2],[4],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[2],[5],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[3],[4],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[3],[5],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[4],[5],[6]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [3,1,4,2] => [3,1,4,2] => 2

Description

The number of stack-sorts needed to sort a permutation. A permutation is (West)

$t$ -stack sortable if it is sortable using

$t$ stacks in series. Let

$W_t(n,k)$ be the number of permutations of size

$n$ with

$k$ descents which are

$t$ -stack sortable. Then the polynomials

$W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have

$W_{n,1}(x) = A_n(x)$ , the Eulerian polynomials. One can show that

$W_{n,1}(x)$ and

$W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern

$231$ have statistic at most

$1$ , see [3]. These are counted by

$\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern

$2341$ and the barred pattern

$3\bar 5241$ have statistic at most

$2$ , see [4]. These are counted by

$\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

Matching statistic: St000159
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [1] => [1]
 => 1
[[1],[3]]
 => [2,1] => [1] => [1]
 => 1
[[2],[3]]
 => [2,1] => [1] => [1]
 => 1
[[1],[4]]
 => [2,1] => [1] => [1]
 => 1
[[2],[4]]
 => [2,1] => [1] => [1]
 => 1
[[3],[4]]
 => [2,1] => [1] => [1]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[5]]
 => [2,1] => [1] => [1]
 => 1
[[2],[5]]
 => [2,1] => [1] => [1]
 => 1
[[3],[5]]
 => [2,1] => [1] => [1]
 => 1
[[4],[5]]
 => [2,1] => [1] => [1]
 => 1
[[1],[2],[4]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[3],[4]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[2],[3],[4]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2
[[1],[6]]
 => [2,1] => [1] => [1]
 => 1
[[2],[6]]
 => [2,1] => [1] => [1]
 => 1
[[3],[6]]
 => [2,1] => [1] => [1]
 => 1
[[4],[6]]
 => [2,1] => [1] => [1]
 => 1
[[5],[6]]
 => [2,1] => [1] => [1]
 => 1
[[1],[2],[5]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[3],[5]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[4],[5]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[2],[3],[5]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[2],[4],[5]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[3],[4],[5]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1] => [3]
 => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [4,2,1,3] => [3,1]
 => 2
[[1],[7]]
 => [2,1] => [1] => [1]
 => 1
[[2],[7]]
 => [2,1] => [1] => [1]
 => 1
[[3],[7]]
 => [2,1] => [1] => [1]
 => 1
[[4],[7]]
 => [2,1] => [1] => [1]
 => 1
[[5],[7]]
 => [2,1] => [1] => [1]
 => 1
[[6],[7]]
 => [2,1] => [1] => [1]
 => 1
[[1],[2],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[3],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[4],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1],[5],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[2],[3],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[2],[4],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[2],[5],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[3],[4],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[3],[5],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[4],[5],[6]]
 => [3,2,1] => [2,1] => [2]
 => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [2,1,3] => [2,1]
 => 2

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St000245

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[3]]
 => [2,1] => [1,2] => [1,2] => 1
[[2],[3]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[4]]
 => [2,1] => [1,2] => [1,2] => 1
[[2],[4]]
 => [2,1] => [1,2] => [1,2] => 1
[[3],[4]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[5]]
 => [2,1] => [1,2] => [1,2] => 1
[[2],[5]]
 => [2,1] => [1,2] => [1,2] => 1
[[3],[5]]
 => [2,1] => [1,2] => [1,2] => 1
[[4],[5]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[2],[4]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[3],[4]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[2],[3],[4]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2
[[1],[6]]
 => [2,1] => [1,2] => [1,2] => 1
[[2],[6]]
 => [2,1] => [1,2] => [1,2] => 1
[[3],[6]]
 => [2,1] => [1,2] => [1,2] => 1
[[4],[6]]
 => [2,1] => [1,2] => [1,2] => 1
[[5],[6]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[2],[5]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[3],[5]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[4],[5]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[2],[3],[5]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[2],[4],[5]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[3],[4],[5]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1,4] => [1,4,3,2] => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,5,1,3,4] => [4,1,3,5,2] => 2
[[1],[7]]
 => [2,1] => [1,2] => [1,2] => 1
[[2],[7]]
 => [2,1] => [1,2] => [1,2] => 1
[[3],[7]]
 => [2,1] => [1,2] => [1,2] => 1
[[4],[7]]
 => [2,1] => [1,2] => [1,2] => 1
[[5],[7]]
 => [2,1] => [1,2] => [1,2] => 1
[[6],[7]]
 => [2,1] => [1,2] => [1,2] => 1
[[1],[2],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[3],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[4],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1],[5],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[2],[3],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[2],[4],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[2],[5],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[3],[4],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[3],[5],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[4],[5],[6]]
 => [3,2,1] => [2,1,3] => [1,3,2] => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [2,4,1,3] => [3,1,2,4] => 2

Description

The number of ascents of a permutation.

Matching statistic: St000260
(load all 6 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1],[3]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[2],[3]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[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],[2],[3]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[5]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[2],[5]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[3],[5]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[4],[5]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[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
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1],[6]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[2],[6]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[3],[6]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[4],[6]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[5],[6]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[[1],[7]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[2],[7]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[3],[7]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[4],[7]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[5],[7]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[6],[7]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000291
(load all 5 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [2,1] => [2] => 10 => 1
[[1],[3]]
 => [2,1] => [2] => 10 => 1
[[2],[3]]
 => [2,1] => [2] => 10 => 1
[[1],[4]]
 => [2,1] => [2] => 10 => 1
[[2],[4]]
 => [2,1] => [2] => 10 => 1
[[3],[4]]
 => [2,1] => [2] => 10 => 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[5]]
 => [2,1] => [2] => 10 => 1
[[2],[5]]
 => [2,1] => [2] => 10 => 1
[[3],[5]]
 => [2,1] => [2] => 10 => 1
[[4],[5]]
 => [2,1] => [2] => 10 => 1
[[1],[2],[4]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[3],[4]]
 => [3,2,1] => [2,1] => 101 => 1
[[2],[3],[4]]
 => [3,2,1] => [2,1] => 101 => 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,2] => 1010 => 2
[[1],[6]]
 => [2,1] => [2] => 10 => 1
[[2],[6]]
 => [2,1] => [2] => 10 => 1
[[3],[6]]
 => [2,1] => [2] => 10 => 1
[[4],[6]]
 => [2,1] => [2] => 10 => 1
[[5],[6]]
 => [2,1] => [2] => 10 => 1
[[1],[2],[5]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[3],[5]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[4],[5]]
 => [3,2,1] => [2,1] => 101 => 1
[[2],[3],[5]]
 => [3,2,1] => [2,1] => 101 => 1
[[2],[4],[5]]
 => [3,2,1] => [2,1] => 101 => 1
[[3],[4],[5]]
 => [3,2,1] => [2,1] => 101 => 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,2] => 1010 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,2] => 1010 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,2] => 1010 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,2] => 1010 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => 1101 => 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,3] => 10100 => 2
[[1],[7]]
 => [2,1] => [2] => 10 => 1
[[2],[7]]
 => [2,1] => [2] => 10 => 1
[[3],[7]]
 => [2,1] => [2] => 10 => 1
[[4],[7]]
 => [2,1] => [2] => 10 => 1
[[5],[7]]
 => [2,1] => [2] => 10 => 1
[[6],[7]]
 => [2,1] => [2] => 10 => 1
[[1],[2],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[3],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[4],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[1],[5],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[2],[3],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[2],[4],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[2],[5],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[3],[4],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[3],[5],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[4],[5],[6]]
 => [3,2,1] => [2,1] => 101 => 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [2,2] => 1010 => 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [2,2] => 1010 => 2

Description

The number of descents of a binary word.

Matching statistic: St000483

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000483: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[1],[3]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[2],[3]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[1],[4]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[2],[4]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[3],[4]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[5]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[2],[5]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[3],[5]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[4],[5]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[1],[6]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[2],[6]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[3],[6]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[4],[6]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[5],[6]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
[[1],[7]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[2],[7]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[3],[7]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[4],[7]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[5],[7]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[6],[7]]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2
[[1,3],[2],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 2

Description

The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]

The following 247 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000537The cutwidth of a graph. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000834The number of right outer peaks of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001096The size of the overlap set of a permutation. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001270The bandwidth of a graph. St001280The number of parts of an integer partition that are at least two. St001644The dimension of a graph. St001737The number of descents of type 2 in a permutation. St001962The proper pathwidth of a graph. St000062The length of the longest increasing subsequence of the permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000542The number of left-to-right-minima of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001642The Prague dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000144The pyramid weight of the Dyck path. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000243The number of cyclic valleys and cyclic peaks of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000486The number of cycles of length at least 3 of a permutation. St000640The rank of the largest boolean interval in a poset. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000308The height of the tree associated to a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000568The hook number of a binary tree. St000630The length of the shortest palindromic decomposition of a binary word. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000701The protection number of a binary tree. St000744The length of the path to the largest entry in a standard Young tableau. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000983The length of the longest alternating subword. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001128The exponens consonantiae of a partition. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001432The order dimension of the partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000292The number of ascents of a binary word. St000353The number of inner valleys of a permutation. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000691The number of changes of a binary word. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000893The number of distinct diagonal sums of an alternating sign matrix. St001569The maximal modular displacement of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001209The pmaj statistic of a parking function. St001864The number of excedances 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. St000633The size of the automorphism group of a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . 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. St000264The girth of a graph, which is not a tree. 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. St000907The number of maximal antichains of minimal length in a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000699The toughness times the least common multiple of 1,. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. 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

$eA$ . St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000679The pruning number of an ordered tree. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001624The breadth of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001926Sparre Andersen's position of the maximum of a signed permutation. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001568The smallest positive integer that does not appear twice in the partition. St000080The rank of the poset. St000284The Plancherel distribution on integer partitions. St000307The number of rowmotion orbits of a poset. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001401The number of distinct entries in a semistandard tableau. St001621The number of atoms of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001877Number of indecomposable injective modules with projective dimension 2. St001407The number of minimal entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001875The number of simple modules with projective dimension at most 1. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001525The number of symmetric hooks on the diagonal of a partition. 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. St001545The second Elser number of a connected graph.