- FindStat

Your data matches 114 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000507
(load all 3 compositions to match this statistic)

Mapping

St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000157
(load all 3 compositions to match this statistic)

Mapping

St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000011

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => [1,0]
 => 1
[[1,2]]
 => 0 => [2] => [1,1,0,0]
 => 1
[[1],[2]]
 => 1 => [1,1] => [1,0,1,0]
 => 2
[[1,2,3]]
 => 00 => [3] => [1,1,1,0,0,0]
 => 1
[[1,3],[2]]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
[[1,2],[3]]
 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000093

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 2
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 3
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 4
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 3
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 5
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 4
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 6
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 5
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 5
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000097

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000288
(load all 46 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => 1 => 1
[[1,2]]
 => 0 => [2] => 10 => 1
[[1],[2]]
 => 1 => [1,1] => 11 => 2
[[1,2,3]]
 => 00 => [3] => 100 => 1
[[1,3],[2]]
 => 10 => [1,2] => 110 => 2
[[1,2],[3]]
 => 01 => [2,1] => 101 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => 111 => 3
[[1,2,3,4]]
 => 000 => [4] => 1000 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => 1100 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => 1010 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => 1001 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => 1101 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => 1010 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => 1110 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => 1101 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => 1011 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => 1111 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => 10000 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => 11000 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => 10100 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => 10010 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => 10001 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => 11010 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => 10100 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => 11001 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => 10101 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => 10010 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => 11100 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => 11010 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => 10110 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => 11001 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => 10101 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => 10011 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => 11101 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => 11010 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => 10110 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => 11011 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => 10101 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => 11110 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => 11101 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => 11011 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => 10111 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => 11111 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => 100000 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => 110000 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => 101000 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => 100100 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => 100010 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => 100001 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => 110100 => 3

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000676

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000786

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 2
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 1
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 3
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 4
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 3
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 5
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 4
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 6
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 5
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 5
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4

Description

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001203We associate to 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 Dyck path as follows: St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000632The jump number of the poset. St001812The biclique partition number of a graph. St000306The bounce count of a Dyck path. St000167The number of leaves of an ordered tree. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001644The dimension of a graph. St000662The staircase size of the code of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001645The pebbling number of a connected graph. St000703The number of deficiencies of a permutation. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral. St000542The number of left-to-right-minima of a permutation. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000083The number of left oriented leafs of a binary tree except the first one. St001480The number of simple summands of the module J^2/J^3. St001674The number of vertices of the largest induced star graph in the graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000159The number of distinct parts of the integer partition. St001321The number of vertices of the largest induced subforest of a graph. St001331The size of the minimal feedback vertex set. St000264The girth of a graph, which is not a tree. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000731The number of double exceedences of a permutation. St000260The radius of a connected graph. St000991The number of right-to-left minima of a permutation. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001727The number of invisible inversions of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001896The number of right descents of a signed permutations. St001060The distinguishing index of a graph. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. 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

$eA$ . St001720The minimal length of a chain of small intervals in a lattice. St000455The second largest eigenvalue of a graph if it is integral.