- FindStat

Your data matches 50 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000288
(load all 14 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => 0 => 0 => 1 => 1
[[1],[2]]
 => 1 => 1 => 0 => 0
[[1,2,3]]
 => 00 => 00 => 11 => 2
[[1,3],[2]]
 => 10 => 01 => 10 => 1
[[1,2],[3]]
 => 01 => 01 => 10 => 1
[[1],[2],[3]]
 => 11 => 11 => 00 => 0
[[1,2,3,4]]
 => 000 => 000 => 111 => 3
[[1,3,4],[2]]
 => 100 => 001 => 110 => 2
[[1,2,4],[3]]
 => 010 => 001 => 110 => 2
[[1,2,3],[4]]
 => 001 => 001 => 110 => 2
[[1,3],[2,4]]
 => 101 => 011 => 100 => 1
[[1,2],[3,4]]
 => 010 => 001 => 110 => 2
[[1,4],[2],[3]]
 => 110 => 011 => 100 => 1
[[1,3],[2],[4]]
 => 101 => 011 => 100 => 1
[[1,2],[3],[4]]
 => 011 => 011 => 100 => 1
[[1],[2],[3],[4]]
 => 111 => 111 => 000 => 0
[[1,2,3,4,5]]
 => 0000 => 0000 => 1111 => 4
[[1,3,4,5],[2]]
 => 1000 => 0001 => 1110 => 3
[[1,2,4,5],[3]]
 => 0100 => 0001 => 1110 => 3
[[1,2,3,5],[4]]
 => 0010 => 0001 => 1110 => 3
[[1,2,3,4],[5]]
 => 0001 => 0001 => 1110 => 3
[[1,3,5],[2,4]]
 => 1010 => 0011 => 1100 => 2
[[1,2,5],[3,4]]
 => 0100 => 0001 => 1110 => 3
[[1,3,4],[2,5]]
 => 1001 => 0011 => 1100 => 2
[[1,2,4],[3,5]]
 => 0101 => 0101 => 1010 => 2
[[1,2,3],[4,5]]
 => 0010 => 0001 => 1110 => 3
[[1,4,5],[2],[3]]
 => 1100 => 0011 => 1100 => 2
[[1,3,5],[2],[4]]
 => 1010 => 0011 => 1100 => 2
[[1,2,5],[3],[4]]
 => 0110 => 0011 => 1100 => 2
[[1,3,4],[2],[5]]
 => 1001 => 0011 => 1100 => 2
[[1,2,4],[3],[5]]
 => 0101 => 0101 => 1010 => 2
[[1,2,3],[4],[5]]
 => 0011 => 0011 => 1100 => 2
[[1,4],[2,5],[3]]
 => 1101 => 0111 => 1000 => 1
[[1,3],[2,5],[4]]
 => 1010 => 0011 => 1100 => 2
[[1,2],[3,5],[4]]
 => 0110 => 0011 => 1100 => 2
[[1,3],[2,4],[5]]
 => 1011 => 0111 => 1000 => 1
[[1,2],[3,4],[5]]
 => 0101 => 0101 => 1010 => 2
[[1,5],[2],[3],[4]]
 => 1110 => 0111 => 1000 => 1
[[1,4],[2],[3],[5]]
 => 1101 => 0111 => 1000 => 1
[[1,3],[2],[4],[5]]
 => 1011 => 0111 => 1000 => 1
[[1,2],[3],[4],[5]]
 => 0111 => 0111 => 1000 => 1
[[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 0000 => 0
[[1,2,3,4,5,6]]
 => 00000 => 00000 => 11111 => 5
[[1,3,4,5,6],[2]]
 => 10000 => 00001 => 11110 => 4
[[1,2,4,5,6],[3]]
 => 01000 => 00001 => 11110 => 4
[[1,2,3,5,6],[4]]
 => 00100 => 00001 => 11110 => 4
[[1,2,3,4,6],[5]]
 => 00010 => 00001 => 11110 => 4
[[1,2,3,4,5],[6]]
 => 00001 => 00001 => 11110 => 4
[[1,3,5,6],[2,4]]
 => 10100 => 00011 => 11100 => 3
[[1,2,5,6],[3,4]]
 => 01000 => 00001 => 11110 => 4

Description

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

Matching statistic: St000507
(load all 4 compositions to match this statistic)

Mapping

St000507: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => 2 = 1 + 1
[[1],[2]]
 => 1 = 0 + 1
[[1,2,3]]
 => 3 = 2 + 1
[[1,3],[2]]
 => 2 = 1 + 1
[[1,2],[3]]
 => 2 = 1 + 1
[[1],[2],[3]]
 => 1 = 0 + 1
[[1,2,3,4]]
 => 4 = 3 + 1
[[1,3,4],[2]]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => 3 = 2 + 1
[[1,3],[2,4]]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => 2 = 1 + 1
[[1,2],[3],[4]]
 => 2 = 1 + 1
[[1],[2],[3],[4]]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => 5 = 4 + 1
[[1,3,4,5],[2]]
 => 4 = 3 + 1
[[1,2,4,5],[3]]
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => 4 = 3 + 1
[[1,2,3,4],[5]]
 => 4 = 3 + 1
[[1,3,5],[2,4]]
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => 4 = 3 + 1
[[1,3,4],[2,5]]
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => 4 = 3 + 1
[[1,4,5],[2],[3]]
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => 3 = 2 + 1
[[1,4],[2,5],[3]]
 => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => 3 = 2 + 1
[[1,2],[3,5],[4]]
 => 3 = 2 + 1
[[1,3],[2,4],[5]]
 => 2 = 1 + 1
[[1,2],[3,4],[5]]
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => 2 = 1 + 1
[[1,3],[2],[4],[5]]
 => 2 = 1 + 1
[[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => 6 = 5 + 1
[[1,3,4,5,6],[2]]
 => 5 = 4 + 1
[[1,2,4,5,6],[3]]
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => 5 = 4 + 1
[[1,2,3,4,5],[6]]
 => 5 = 4 + 1
[[1,3,5,6],[2,4]]
 => 4 = 3 + 1
[[1,2,5,6],[3,4]]
 => 5 = 4 + 1
[[1,2,3,4,5,6,7,9],[8]]
 => ? = 7 + 1
[[1,2,3,4,5,6,9],[7,8]]
 => ? = 7 + 1
[[1,2,3,4,5,6,9],[7],[8]]
 => ? = 6 + 1
[[1,2,3,4,5,9],[6,7,8]]
 => ? = 7 + 1
[[1,2,3,4,5,9],[6,7],[8]]
 => ? = 6 + 1
[[1,2,3,4,5,9],[6],[7],[8]]
 => ? = 5 + 1
[[1,2,3,4,9],[5,6,7],[8]]
 => ? = 6 + 1
[[1,2,3,4,9],[5,6],[7],[8]]
 => ? = 5 + 1
[[1,2,3,4,9],[5],[6],[7],[8]]
 => ? = 4 + 1
[[1,2,3,7],[4,5,6,9],[8]]
 => ? = 6 + 1
[[1,2,3,9],[4,5,6],[7,8]]
 => ? = 6 + 1
[[1,2,3,9],[4,5,6],[7],[8]]
 => ? = 5 + 1
[[1,2,3,9],[4,5],[6,7],[8]]
 => ? = 5 + 1
[[1,2,3,9],[4,5],[6],[7],[8]]
 => ? = 4 + 1
[[1,2,3,9],[4],[5],[6],[7],[8]]
 => ? = 3 + 1
[[1,2,5],[3,4,8],[6,7,9]]
 => ? = 5 + 1
[[1,2,5],[3,4,9],[6,7],[8]]
 => ? = 5 + 1
[[1,2,5],[3,4,9],[6],[7],[8]]
 => ? = 4 + 1
[[1,2,9],[3,4],[5,6],[7],[8]]
 => ? = 4 + 1
[[1,2,9],[3,4],[5],[6],[7],[8]]
 => ? = 3 + 1
[[1,3],[2,5],[4,9],[6],[7],[8]]
 => ? = 3 + 1
[[1,3],[2,9],[4],[5],[6],[7],[8]]
 => ? = 2 + 1
[[1,2,3,4,5,6,7,10],[8,9]]
 => ? = 8 + 1
[[1,2,3,4,5,6,7,10],[8],[9]]
 => ? = 7 + 1
[[1,2,3,4,5,6,10],[7,8],[9]]
 => ? = 7 + 1
[[1,2,3,4,5,6,10],[7],[8],[9]]
 => ? = 6 + 1
[[1,2,3,4,5,10],[6,7,8,9]]
 => ? = 8 + 1
[[1,2,3,4,5,10],[6,7],[8,9]]
 => ? = 7 + 1
[[1,2,3,4,5,10],[6,7],[8],[9]]
 => ? = 6 + 1
[[1,2,3,4,5,10],[6],[7],[8],[9]]
 => ? = 5 + 1
[[1,2,3,4,10],[5,6,7,8],[9]]
 => ? = 7 + 1
[[1,2,3,4,10],[5,6],[7,8],[9]]
 => ? = 6 + 1
[[1,2,3,4,10],[5,6],[7],[8],[9]]
 => ? = 5 + 1
[[1,2,3,4,10],[5],[6],[7],[8],[9]]
 => ? = 4 + 1
[[1,2,3,10],[4,5],[6,7],[8,9]]
 => ? = 6 + 1
[[1,2,3,10],[4,5],[6,7],[8],[9]]
 => ? = 5 + 1
[[1,2,3,10],[4,5],[6],[7],[8],[9]]
 => ? = 4 + 1
[[1,2,3,10],[4],[5],[6],[7],[8],[9]]
 => ? = 3 + 1
[[1,2,5],[3,4,10],[6,7],[8,9]]
 => ? = 6 + 1
[[1,2,5],[3,4,10],[6,7],[8],[9]]
 => ? = 5 + 1
[[1,2,10],[3,4],[5,6],[7,8],[9]]
 => ? = 5 + 1
[[1,2,10],[3,4],[5,6],[7],[8],[9]]
 => ? = 4 + 1
[[1,2,10],[3,4],[5],[6],[7],[8],[9]]
 => ? = 3 + 1
[[1,3],[2,5],[4,7],[6,10],[8],[9]]
 => ? = 4 + 1
[[1,3],[2,5],[4,10],[6],[7],[8],[9]]
 => ? = 3 + 1
[[1,3],[2,10],[4],[5],[6],[7],[8],[9]]
 => ? = 2 + 1
[[1,3,4,6,7,8,9],[2,5]]
 => ? = 6 + 1
[[1,3,5,6,7,8,9],[2],[4]]
 => ? = 6 + 1
[[1,3,5,6,8,9],[2,4,7]]
 => ? = 5 + 1
[[1,2,4,7,8,9],[3,5],[6]]
 => ? = 5 + 1

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 2 compositions to match this statistic)

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [[1],[2]]
 => [[1],[2]]
 => 1
[[1],[2]]
 => [[1,2]]
 => [[1,2]]
 => 0
[[1,2,3]]
 => [[1],[2],[3]]
 => [[1],[2],[3]]
 => 2
[[1,3],[2]]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[[1],[2],[3]]
 => [[1,2,3]]
 => [[1,2,3]]
 => 0
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => 3
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 2
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => [[1,3],[2],[4]]
 => 2
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 2
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [[1,2],[3,4]]
 => 1
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 2
[[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 1
[[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 1
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [[1,2,3,4]]
 => 0
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => 4
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => [[1,4],[2],[3],[5]]
 => 3
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => [[1,3],[2],[4],[5]]
 => 3
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => 3
[[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [[1,3],[2,5],[4]]
 => 2
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [[1,4],[2,5],[3]]
 => 3
[[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => [[1,2],[3,5],[4]]
 => 2
[[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => [[1,2],[3,4],[5]]
 => 2
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => [[1,3],[2,4],[5]]
 => 3
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 2
[[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => [[1,3,4],[2],[5]]
 => 2
[[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => [[1,2,5],[3],[4]]
 => 2
[[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 2
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 2
[[1,4],[2,5],[3]]
 => [[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => 1
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 2
[[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => [[1,3,4],[2,5]]
 => 2
[[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => 1
[[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => 2
[[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 1
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 1
[[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => 1
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 0
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 4
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => [[1,5],[2],[3],[4],[6]]
 => 4
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => [[1,4],[2],[3],[5],[6]]
 => 4
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => [[1,3],[2],[4],[5],[6]]
 => 4
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => 4
[[1,3,5,6],[2,4]]
 => [[1,2],[3,4],[5],[6]]
 => [[1,4],[2,6],[3],[5]]
 => 3
[[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => [[1,5],[2,6],[3],[4]]
 => 4
[[1,8],[2,9],[3],[4],[5],[6],[7]]
 => ?
 => ?
 => ? = 1
[[1,7],[2,8],[3,9],[4],[5],[6]]
 => ?
 => ?
 => ? = 1
[[1,7,9],[2,8],[3],[4],[5],[6]]
 => ?
 => ?
 => ? = 2
[[1,6,9],[2,7],[3,8],[4],[5]]
 => ?
 => ?
 => ? = 2
[[1,6,8],[2,7,9],[3],[4],[5]]
 => ?
 => ?
 => ? = 2
[[1,5,8],[2,6,9],[3,7],[4]]
 => ?
 => ?
 => ? = 2
[[1,5,7,9],[2,6,8],[3],[4]]
 => ?
 => ?
 => ? = 3
[[1,4,7,9],[2,5,8],[3,6]]
 => ?
 => ?
 => ? = 3
[[1,4,6,8,9],[2,5,7],[3]]
 => ?
 => ?
 => ? = 4
[[1,8],[2,9],[3,10],[4],[5],[6],[7]]
 => ?
 => ?
 => ? = 1
[[1,8,10],[2,9],[3],[4],[5],[6],[7]]
 => ?
 => ?
 => ? = 2
[[1,7,10],[2,8],[3,9],[4],[5],[6]]
 => ?
 => ?
 => ? = 2
[[1,6,10],[2,7],[3,8],[4,9],[5]]
 => ?
 => ?
 => ? = 2
[[1,6,9],[2,7,10],[3,8],[4],[5]]
 => ?
 => ?
 => ? = 2
[[1,6,8,10],[2,7,9],[3],[4],[5]]
 => ?
 => ?
 => ? = 3
[[1,5,8,10],[2,6,9],[3,7],[4]]
 => ?
 => ?
 => ? = 3
[[1,4,7,9],[2,5,8,10],[3,6]]
 => ?
 => ?
 => ? = 3
[[1,4,7,9,10],[2,5,8],[3,6]]
 => ?
 => ?
 => ? = 4
[[1,3],[2,4],[5],[6],[7],[8],[9]]
 => ?
 => ?
 => ? = 1
[[1,4],[2,5],[3,6],[7],[8],[9]]
 => ?
 => ?
 => ? = 1
[[1,5],[2,6],[3,7],[4,8],[9]]
 => ?
 => ?
 => ? = 1
[[1,3,5],[2,4,6],[7],[8],[9]]
 => ?
 => ?
 => ? = 2
[[1,3,6],[2,4,7],[5,8],[9]]
 => ?
 => ?
 => ? = 2
[[1,3,5,7],[2,4,6,8],[9]]
 => ?
 => ?
 => ? = 3
[[1,4],[2,5],[3,6],[7],[8],[9],[10]]
 => ?
 => ?
 => ? = 1
[[1,3,6],[2,4,7],[5,8],[9],[10]]
 => ?
 => ?
 => ? = 2
[[1,4,7],[2,5,8],[3,6,9],[10]]
 => ?
 => ?
 => ? = 2
[[1,3,5,8],[2,4,6,9],[7,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,4],[5,6],[7,9],[8,10]]
 => ?
 => ?
 => ? = 4
[[1,2],[3,4],[5,7],[6,8],[9,10]]
 => ?
 => ?
 => ? = 4
[[1,2],[3,4],[5,7],[6,9],[8,10]]
 => ?
 => ?
 => ? = 4
[[1,2],[3,4],[5,8],[6,9],[7,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,5],[4,6],[7,8],[9,10]]
 => ?
 => ?
 => ? = 4
[[1,2],[3,5],[4,6],[7,9],[8,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,5],[4,7],[6,8],[9,10]]
 => ?
 => ?
 => ? = 4
[[1,2],[3,5],[4,7],[6,9],[8,10]]
 => ?
 => ?
 => ? = 4
[[1,2],[3,5],[4,8],[6,9],[7,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,6],[4,7],[5,8],[9,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,6],[4,7],[5,9],[8,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,6],[4,8],[5,9],[7,10]]
 => ?
 => ?
 => ? = 3
[[1,2],[3,7],[4,8],[5,9],[6,10]]
 => ?
 => ?
 => ? = 2
[[1,3],[2,4],[5,6],[7,8],[9,10]]
 => ?
 => ?
 => ? = 4
[[1,3],[2,4],[5,6],[7,9],[8,10]]
 => ?
 => ?
 => ? = 3
[[1,3],[2,4],[5,7],[6,8],[9,10]]
 => ?
 => ?
 => ? = 3
[[1,3],[2,4],[5,7],[6,9],[8,10]]
 => ?
 => ?
 => ? = 3
[[1,3],[2,4],[5,8],[6,9],[7,10]]
 => ?
 => ?
 => ? = 2
[[1,3],[2,5],[4,6],[7,8],[9,10]]
 => ?
 => ?
 => ? = 4
[[1,3],[2,5],[4,6],[7,9],[8,10]]
 => ?
 => ?
 => ? = 3
[[1,3],[2,5],[4,7],[6,8],[9,10]]
 => ?
 => ?
 => ? = 4
[[1,3],[2,5],[4,7],[6,9],[8,10]]
 => ?
 => ?
 => ? = 4

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

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [1,2] => [1,2] => [1,1]
 => 2 = 1 + 1
[[1],[2]]
 => [2,1] => [2,1] => [2]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [2,3,1] => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [3]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => [2,1,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,3,4,1] => [2,1,1]
 => 3 = 2 + 1
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => [2,2]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => [2,1,1]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [3,1]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,2,4,1] => [3,1]
 => 2 = 1 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => [3,1]
 => 2 = 1 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [4]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5 = 4 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
 => 4 = 3 + 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => [2,1,1,1]
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => [2,1,1,1]
 => 4 = 3 + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,3,4,5,1] => [2,1,1,1]
 => 4 = 3 + 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [3,1,4,2,5] => [2,2,1]
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => [2,1,1,1]
 => 4 = 3 + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [3,1,4,5,2] => [2,2,1]
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,4,1,5,2] => [2,2,1]
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,4,5,1,2] => [2,1,1,1]
 => 4 = 3 + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,1,1]
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,2,4,1,5] => [3,1,1]
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,4,2,1,5] => [3,1,1]
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,2,4,5,1] => [3,1,1]
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,4,2,5,1] => [3,1,1]
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,2,1,5,3] => [3,2]
 => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => [3,1,1]
 => 3 = 2 + 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,2,1,3] => [3,1,1]
 => 3 = 2 + 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [4,2,5,3,1] => [3,2]
 => 2 = 1 + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [4,5,2,3,1] => [3,1,1]
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,2,5,1] => [4,1]
 => 2 = 1 + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,3,5,2,1] => [4,1]
 => 2 = 1 + 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [4,5,3,2,1] => [4,1]
 => 2 = 1 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [5]
 => 1 = 0 + 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 = 5 + 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,1,1,1]
 => 5 = 4 + 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [2,3,1,4,5,6] => [2,1,1,1,1]
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [2,3,4,1,5,6] => [2,1,1,1,1]
 => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => [2,1,1,1,1]
 => 5 = 4 + 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 5 = 4 + 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [3,1,4,2,5,6] => [2,2,1,1]
 => 4 = 3 + 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,4,1,2,5,6] => [2,1,1,1,1]
 => 5 = 4 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,6,1,3,4,5,7,8] => [3,1,4,5,6,2,7,8] => ?
 => ? = 5 + 1
[[1,2,3,4,7,8],[5],[6]]
 => [6,5,1,2,3,4,7,8] => [3,4,5,6,2,1,7,8] => ?
 => ? = 5 + 1
[[1,2,3,5,6,8],[4],[7]]
 => [7,4,1,2,3,5,6,8] => [3,4,5,2,6,7,1,8] => ?
 => ? = 5 + 1
[[1,2,5,7,8],[3,4,6]]
 => [3,4,6,1,2,5,7,8] => [4,5,1,2,6,3,7,8] => ?
 => ? = 5 + 1
[[1,3,5,6,8],[2,4,7]]
 => [2,4,7,1,3,5,6,8] => [4,1,5,2,6,7,3,8] => ?
 => ? = 4 + 1
[[1,3,4,6,8],[2,5,7]]
 => [2,5,7,1,3,4,6,8] => [4,1,5,6,2,7,3,8] => ?
 => ? = 4 + 1
[[1,3,4,6,7],[2,5,8]]
 => [2,5,8,1,3,4,6,7] => [4,1,5,6,2,7,8,3] => ?
 => ? = 4 + 1
[[1,2,5,7,8],[3,6],[4]]
 => [4,3,6,1,2,5,7,8] => [4,5,2,1,6,3,7,8] => ?
 => ? = 4 + 1
[[1,3,4,7,8],[2,6],[5]]
 => [5,2,6,1,3,4,7,8] => [4,2,5,6,1,3,7,8] => ?
 => ? = 5 + 1
[[1,2,3,7,8],[4,6],[5]]
 => [5,4,6,1,2,3,7,8] => [4,5,6,2,1,3,7,8] => ?
 => ? = 5 + 1
[[1,3,5,7,8],[2,4],[6]]
 => [6,2,4,1,3,5,7,8] => [4,2,5,3,6,1,7,8] => ?
 => ? = 4 + 1
[[1,2,5,7,8],[3,4],[6]]
 => [6,3,4,1,2,5,7,8] => [4,5,2,3,6,1,7,8] => ?
 => ? = 5 + 1
[[1,3,4,7,8],[2,5],[6]]
 => [6,2,5,1,3,4,7,8] => [4,2,5,6,3,1,7,8] => ?
 => ? = 4 + 1
[[1,2,3,7,8],[4,5],[6]]
 => [6,4,5,1,2,3,7,8] => [4,5,6,2,3,1,7,8] => ?
 => ? = 5 + 1
[[1,3,5,6,8],[2,7],[4]]
 => [4,2,7,1,3,5,6,8] => [4,2,5,1,6,7,3,8] => ?
 => ? = 4 + 1
[[1,3,5,6,8],[2,4],[7]]
 => [7,2,4,1,3,5,6,8] => [4,2,5,3,6,7,1,8] => ?
 => ? = 4 + 1
[[1,3,4,6,8],[2,5],[7]]
 => [7,2,5,1,3,4,6,8] => [4,2,5,6,3,7,1,8] => ?
 => ? = 4 + 1
[[1,2,3,6,8],[4,5],[7]]
 => [7,4,5,1,2,3,6,8] => [4,5,6,2,3,7,1,8] => ?
 => ? = 5 + 1
[[1,3,4,5,8],[2,6],[7]]
 => [7,2,6,1,3,4,5,8] => [4,2,5,6,7,3,1,8] => ?
 => ? = 4 + 1
[[1,2,4,6,7],[3,8],[5]]
 => [5,3,8,1,2,4,6,7] => [4,5,2,6,1,7,8,3] => ?
 => ? = 4 + 1
[[1,2,3,5,6],[4,8],[7]]
 => [7,4,8,1,2,3,5,6] => [4,5,6,2,7,8,1,3] => ?
 => ? = 5 + 1
[[1,3,4,6,7],[2,5],[8]]
 => [8,2,5,1,3,4,6,7] => [4,2,5,6,3,7,8,1] => ?
 => ? = 4 + 1
[[1,2,4,5,6],[3,7],[8]]
 => [8,3,7,1,2,4,5,6] => [4,5,2,6,7,8,3,1] => ?
 => ? = 4 + 1
[[1,3,4,7,8],[2],[5],[6]]
 => [6,5,2,1,3,4,7,8] => [4,3,5,6,2,1,7,8] => ?
 => ? = 4 + 1
[[1,2,4,7,8],[3],[5],[6]]
 => [6,5,3,1,2,4,7,8] => [4,5,3,6,2,1,7,8] => ?
 => ? = 4 + 1
[[1,3,5,6,8],[2],[4],[7]]
 => [7,4,2,1,3,5,6,8] => [4,3,5,2,6,7,1,8] => ?
 => ? = 4 + 1
[[1,2,5,6,8],[3],[4],[7]]
 => [7,4,3,1,2,5,6,8] => [4,5,3,2,6,7,1,8] => ?
 => ? = 4 + 1
[[1,2,3,6,8],[4],[5],[7]]
 => [7,5,4,1,2,3,6,8] => [4,5,6,3,2,7,1,8] => ?
 => ? = 4 + 1
[[1,3,4,5,8],[2],[6],[7]]
 => [7,6,2,1,3,4,5,8] => [4,3,5,6,7,2,1,8] => ?
 => ? = 4 + 1
[[1,2,4,5,8],[3],[6],[7]]
 => [7,6,3,1,2,4,5,8] => [4,5,3,6,7,2,1,8] => ?
 => ? = 4 + 1
[[1,2,3,5,8],[4],[6],[7]]
 => [7,6,4,1,2,3,5,8] => [4,5,6,3,7,2,1,8] => ?
 => ? = 4 + 1
[[1,2,4,6,7],[3],[5],[8]]
 => [8,5,3,1,2,4,6,7] => [4,5,3,6,2,7,8,1] => ?
 => ? = 4 + 1
[[1,3,4,5,7],[2],[6],[8]]
 => [8,6,2,1,3,4,5,7] => [4,3,5,6,7,2,8,1] => ?
 => ? = 4 + 1
[[1,2,4,5,6],[3],[7],[8]]
 => [8,7,3,1,2,4,5,6] => [4,5,3,6,7,8,2,1] => ?
 => ? = 4 + 1
[[1,2,3,5,6],[4],[7],[8]]
 => [8,7,4,1,2,3,5,6] => [4,5,6,3,7,8,2,1] => ?
 => ? = 4 + 1
[[1,4,5,8],[2,6,7],[3]]
 => [3,2,6,7,1,4,5,8] => [5,2,1,6,7,3,4,8] => ?
 => ? = 4 + 1
[[1,2,5,8],[3,6,7],[4]]
 => [4,3,6,7,1,2,5,8] => [5,6,2,1,7,3,4,8] => ?
 => ? = 4 + 1
[[1,3,5,8],[2,4,7],[6]]
 => [6,2,4,7,1,3,5,8] => [5,2,6,3,7,1,4,8] => ?
 => ? = 4 + 1
[[1,2,5,8],[3,4,7],[6]]
 => [6,3,4,7,1,2,5,8] => [5,6,2,3,7,1,4,8] => ?
 => ? = 5 + 1
[[1,3,4,8],[2,5,7],[6]]
 => [6,2,5,7,1,3,4,8] => [5,2,6,7,3,1,4,8] => ?
 => ? = 4 + 1
[[1,3,4,8],[2,5,6],[7]]
 => [7,2,5,6,1,3,4,8] => [5,2,6,7,3,4,1,8] => ?
 => ? = 4 + 1
[[1,4,6,7],[2,5,8],[3]]
 => [3,2,5,8,1,4,6,7] => [5,2,1,6,3,7,8,4] => ?
 => ? = 3 + 1
[[1,2,6,7],[3,5,8],[4]]
 => [4,3,5,8,1,2,6,7] => [5,6,2,1,3,7,8,4] => ?
 => ? = 4 + 1
[[1,4,5,7],[2,6,8],[3]]
 => [3,2,6,8,1,4,5,7] => [5,2,1,6,7,3,8,4] => ?
 => ? = 3 + 1
[[1,3,4,7],[2,6,8],[5]]
 => [5,2,6,8,1,3,4,7] => [5,2,6,7,1,3,8,4] => ?
 => ? = 4 + 1
[[1,2,5,7],[3,4,8],[6]]
 => [6,3,4,8,1,2,5,7] => [5,6,2,3,7,1,8,4] => ?
 => ? = 4 + 1
[[1,3,4,7],[2,5,8],[6]]
 => [6,2,5,8,1,3,4,7] => [5,2,6,7,3,1,8,4] => ?
 => ? = 3 + 1
[[1,3,5,6],[2,7,8],[4]]
 => [4,2,7,8,1,3,5,6] => [5,2,6,1,7,8,3,4] => ?
 => ? = 4 + 1
[[1,3,4,6],[2,7,8],[5]]
 => [5,2,7,8,1,3,4,6] => [5,2,6,7,1,8,3,4] => ?
 => ? = 4 + 1
[[1,2,4,5],[3,7,8],[6]]
 => [6,3,7,8,1,2,4,5] => [5,6,2,7,8,1,3,4] => ?
 => ? = 5 + 1

Description

The length of the partition.

Matching statistic: St000394

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 91%

Values

[[1,2]]
 => 0 => [2] => [1,1,0,0]
 => 1
[[1],[2]]
 => 1 => [1,1] => [1,0,1,0]
 => 0
[[1,2,3]]
 => 00 => [3] => [1,1,1,0,0,0]
 => 2
[[1,3],[2]]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[[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]
 => 1
[[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]
 => 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,5,6],[3,4]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,6,7],[5,8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,7],[6,8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,6,7],[5],[8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,7],[6],[8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,6,8],[4,5,7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,5,8],[4,6,7]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,5,7],[4,6,8]]
 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,4,7],[5,6,8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,5,6],[4,7,8]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,7,8],[4,6],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,3,7,8],[4,5],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,6,8],[4,7],[5]]
 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[[1,2,3,5,8],[4,7],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,4,8],[5,7],[6]]
 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,6,8],[4,5],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,5,8],[4,6],[7]]
 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4
[[1,2,3,6,7],[4,8],[5]]
 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[[1,2,3,5,7],[4,8],[6]]
 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,4,7],[5,8],[6]]
 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,5,6],[4,8],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,5],[6,8],[7]]
 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
[[1,2,3,5,7],[4,6],[8]]
 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,4,7],[5,6],[8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,5,6],[4,7],[8]]
 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 4
[[1,2,3,4,6],[5,7],[8]]
 => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[1,2,3,4,5],[6,7],[8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,7,8],[4],[5],[6]]
 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,3,6,8],[4],[5],[7]]
 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[[1,2,3,5,8],[4],[6],[7]]
 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4
[[1,2,3,4,8],[5],[6],[7]]
 => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[1,2,3,6,7],[4],[5],[8]]
 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[[1,2,3,5,7],[4],[6],[8]]
 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,4,7],[5],[6],[8]]
 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,5,6],[4],[7],[8]]
 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 4
[[1,2,3,4,6],[5],[7],[8]]
 => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[1,2,3,6],[4,5,7,8]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,5],[4,6,7,8]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,8],[4,6,7],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,3,8],[4,5,7],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,8],[4,5,6],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,7],[4,6,8],[5]]
 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[[1,2,3,7],[4,5,8],[6]]
 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,6],[4,7,8],[5]]
 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

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: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 64%

Values

[[1,2]]
 => 0 => [2] => ([],2)
 => 2 = 1 + 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 3 = 2 + 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 4 = 3 + 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 5 = 4 + 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[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 = 2 + 1
[[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 + 1
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[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 + 1
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 = 0 + 1
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 6 = 5 + 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 5 = 4 + 1
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1,2,5,6],[3,4]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,4,5,6,7,8]]
 => 0000000 => [8] => ([],8)
 => ? = 7 + 1
[[1,3,4,5,6,7,8],[2]]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 6 + 1
[[1,2,4,5,6,7,8],[3]]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,5,6,7,8],[4]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,6,7,8],[5]]
 => 0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,5,7,8],[6]]
 => 0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,5,6,8],[7]]
 => 0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,5,6,7,8],[2,4]]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,5,6,7,8],[3,4]]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,6,7,8],[2,5]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,6,7,8],[3,5]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,6,7,8],[4,5]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,5,7,8],[2,6]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,7,8],[3,6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,7,8],[5,6]]
 => 0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,5,6,8],[2,7]]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,6,8],[3,7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,6,8],[5,7]]
 => 0001010 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,5,8],[6,7]]
 => 0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,5,6],[7,8]]
 => 0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,4,5,6,7,8],[2],[3]]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,5,6,7,8],[2],[4]]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,5,6,7,8],[3],[4]]
 => 0110000 => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,7,8],[2],[5]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,6,7,8],[3],[5]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,7,8],[2],[6]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,7,8],[3],[6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,6,8],[2],[7]]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,6,8],[3],[7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,6,8],[5],[7]]
 => 0001010 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,5,7,8],[2,4,6]]
 => 1010100 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,5,7,8],[3,4,6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,7,8],[2,5,6]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,7,8],[3,5,6]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,7,8],[4,5,6]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,5,6,8],[2,4,7]]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,5,6,8],[3,4,7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,4,6,8],[3,5,7]]
 => 0101010 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,3,6,8],[4,5,7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,8],[3,6,7]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,8],[4,6,7]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1

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: 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: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 64%

Values

[[1,2]]
 => 0 => [2] => ([],2)
 => 2 = 1 + 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 3 = 2 + 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 4 = 3 + 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 5 = 4 + 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 4 = 3 + 1
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[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 = 2 + 1
[[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 + 1
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[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 + 1
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 + 1
[[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 = 0 + 1
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 6 = 5 + 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 5 = 4 + 1
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1,2,5,6],[3,4]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 5 = 4 + 1
[[1,2,3,4,5,6,7,8]]
 => 0000000 => [8] => ([],8)
 => ? = 7 + 1
[[1,3,4,5,6,7,8],[2]]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 6 + 1
[[1,2,4,5,6,7,8],[3]]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,5,6,7,8],[4]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,6,7,8],[5]]
 => 0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,5,7,8],[6]]
 => 0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,5,6,8],[7]]
 => 0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,5,6,7,8],[2,4]]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,5,6,7,8],[3,4]]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,6,7,8],[2,5]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,6,7,8],[3,5]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,6,7,8],[4,5]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,5,7,8],[2,6]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,7,8],[3,6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,7,8],[5,6]]
 => 0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,4,5,6,8],[2,7]]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,6,8],[3,7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,6,8],[5,7]]
 => 0001010 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,5,8],[6,7]]
 => 0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,2,3,4,5,6],[7,8]]
 => 0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,4,5,6,7,8],[2],[3]]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,5,6,7,8],[2],[4]]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,5,6,7,8],[3],[4]]
 => 0110000 => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,7,8],[2],[5]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,6,7,8],[3],[5]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,7,8],[2],[6]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,7,8],[3],[6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,6,8],[2],[7]]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,6,8],[3],[7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,6,8],[5],[7]]
 => 0001010 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,5,7,8],[2,4,6]]
 => 1010100 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,5,7,8],[3,4,6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,7,8],[2,5,6]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,7,8],[3,5,6]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,7,8],[4,5,6]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 6 + 1
[[1,3,5,6,8],[2,4,7]]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,5,6,8],[3,4,7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,6,8],[2,5,7]]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,4,6,8],[3,5,7]]
 => 0101010 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[[1,2,3,6,8],[4,5,7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,3,4,5,8],[2,6,7]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,4,5,8],[3,6,7]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[[1,2,3,5,8],[4,6,7]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1

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

$3$ .

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: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 73%

Values

[[1,2]]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 2 = 1 + 1
[[1],[2]]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[1,3],[2]]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[1,3],[2,4]]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 3 + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 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 = 5 + 1
[[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 = 4 + 1
[[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 = 4 + 1
[[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 = 4 + 1
[[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 = 4 + 1
[[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 = 4 + 1
[[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 = 3 + 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[[1,2,4,5,6,7,8],[3]]
 => [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[[1,2,3,5,6,7,8],[4]]
 => [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[[1,2,3,4,6,7,8],[5]]
 => [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[[1,2,3,4,5,7,8],[6]]
 => [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[[1,2,3,4,5,6,8],[7]]
 => [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[[1,3,5,6,7,8],[2,4]]
 => [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[[1,3,4,6,7,8],[2,5]]
 => [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,4,6,7,8],[3,5]]
 => [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,3,6,7,8],[4,5]]
 => [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[[1,3,4,5,7,8],[2,6]]
 => [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,4,5,7,8],[3,6]]
 => [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,3,5,7,8],[4,6]]
 => [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,3,4,7,8],[5,6]]
 => [5,6,1,2,3,4,7,8] => ?
 => ?
 => ? = 6 + 1
[[1,3,4,5,6,8],[2,7]]
 => [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,2,4,5,6,8],[3,7]]
 => [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,2,3,5,6,8],[4,7]]
 => [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
 => [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,2,3,4,6,8],[5,7]]
 => [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => ? = 5 + 1
[[1,2,3,4,5,8],[6,7]]
 => [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[[1,3,4,5,6,7],[2,8]]
 => [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,4,5,6,7],[3,8]]
 => [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,3,5,6,7],[4,8]]
 => [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,3,4,6,7],[5,8]]
 => [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 5 + 1
[[1,2,3,4,5,7],[6,8]]
 => [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,3,5,6,7,8],[2],[4]]
 => [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[[1,2,5,6,7,8],[3],[4]]
 => [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,3,4,6,7,8],[2],[5]]
 => [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,4,6,7,8],[3],[5]]
 => [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => ? = 5 + 1
[[1,2,3,6,7,8],[4],[5]]
 => [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[[1,3,4,5,7,8],[2],[6]]
 => [6,2,1,3,4,5,7,8] => ?
 => ?
 => ? = 5 + 1
[[1,2,4,5,7,8],[3],[6]]
 => [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,3,5,7,8],[4],[6]]
 => [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,2,3,4,7,8],[5],[6]]
 => [6,5,1,2,3,4,7,8] => ?
 => ?
 => ? = 5 + 1
[[1,3,4,5,6,8],[2],[7]]
 => [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,4,5,6,8],[3],[7]]
 => [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,3,5,6,8],[4],[7]]
 => [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,3,4,6,8],[5],[7]]
 => [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,2,3,4,5,8],[6],[7]]
 => [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,3,4,5,6,7],[2],[8]]
 => [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,4,5,6,7],[3],[8]]
 => [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,3,5,6,7],[4],[8]]
 => [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[[1,2,3,4,6,7],[5],[8]]
 => [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[[1,2,3,4,5,7],[6],[8]]
 => [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
[[1,3,5,7,8],[2,4,6]]
 => [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 4 + 1
[[1,2,5,7,8],[3,4,6]]
 => [3,4,6,1,2,5,7,8] => ?
 => ?
 => ? = 5 + 1
[[1,3,4,7,8],[2,5,6]]
 => [2,5,6,1,3,4,7,8] => ?
 => ?
 => ? = 5 + 1
[[1,2,4,7,8],[3,5,6]]
 => [3,5,6,1,2,4,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
 => ? = 5 + 1

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: St000024
(load all 2 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 64%

Values

[[1,2]]
 => 0 => [2] => [1,1,0,0]
 => 1
[[1],[2]]
 => 1 => [1,1] => [1,0,1,0]
 => 0
[[1,2,3]]
 => 00 => [3] => [1,1,1,0,0,0]
 => 2
[[1,3],[2]]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[[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]
 => 1
[[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]
 => 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,5,6],[3,4]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,4,5,6,7,8]]
 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[[1,3,4,5,6,7,8],[2]]
 => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[[1,2,4,5,6,7,8],[3]]
 => 0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[[1,2,3,5,6,7,8],[4]]
 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 6
[[1,2,3,4,6,7,8],[5]]
 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 6
[[1,2,3,4,5,7,8],[6]]
 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 6
[[1,2,3,4,5,6,8],[7]]
 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[[1,2,3,4,5,6,7],[8]]
 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[[1,3,5,6,7,8],[2,4]]
 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[[1,2,5,6,7,8],[3,4]]
 => 0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
[[1,3,4,6,7,8],[2,5]]
 => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,4,6,7,8],[3,5]]
 => 0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,3,6,7,8],[4,5]]
 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 6
[[1,3,4,5,7,8],[2,6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,4,5,7,8],[3,6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,4,7,8],[5,6]]
 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 6
[[1,3,4,5,6,8],[2,7]]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,4,5,6,8],[3,7]]
 => 0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,6,8],[5,7]]
 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,5,8],[6,7]]
 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 6
[[1,3,4,5,6,7],[2,8]]
 => 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[[1,2,4,5,6,7],[3,8]]
 => 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 5
[[1,2,3,5,6,7],[4,8]]
 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[[1,2,3,4,6,7],[5,8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,7],[6,8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,6],[7,8]]
 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[[1,4,5,6,7,8],[2],[3]]
 => 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5
[[1,3,5,6,7,8],[2],[4]]
 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[[1,2,5,6,7,8],[3],[4]]
 => 0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[[1,3,4,6,7,8],[2],[5]]
 => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,4,6,7,8],[3],[5]]
 => 0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 5
[[1,3,4,5,7,8],[2],[6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,4,5,7,8],[3],[6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 5
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 5
[[1,3,4,5,6,8],[2],[7]]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,4,5,6,8],[3],[7]]
 => 0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,6,8],[5],[7]]
 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
[[1,3,4,5,6,7],[2],[8]]
 => 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[[1,2,4,5,6,7],[3],[8]]
 => 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 5
[[1,2,3,5,6,7],[4],[8]]
 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[[1,2,3,4,6,7],[5],[8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,7],[6],[8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,6],[7],[8]]
 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5
[[1,3,5,7,8],[2,4,6]]
 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000053

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 64%

Values

[[1,2]]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1
[[1],[2]]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 0
[[1,2,3]]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 2
[[1,3],[2]]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[1,2],[3]]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 0
[[1,2,3,4]]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2]]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 4
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[[1,2,4,5,6,7,8],[3]]
 => [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[[1,2,3,5,6,7,8],[4]]
 => [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[[1,2,3,4,6,7,8],[5]]
 => [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6
[[1,2,3,4,5,7,8],[6]]
 => [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6
[[1,2,3,4,5,6,8],[7]]
 => [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6
[[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6
[[1,3,5,6,7,8],[2,4]]
 => [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[[1,3,4,6,7,8],[2,5]]
 => [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,4,6,7,8],[3,5]]
 => [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,3,6,7,8],[4,5]]
 => [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[[1,3,4,5,7,8],[2,6]]
 => [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,4,5,7,8],[3,6]]
 => [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,3,5,7,8],[4,6]]
 => [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,3,4,7,8],[5,6]]
 => [5,6,1,2,3,4,7,8] => ?
 => ?
 => ? = 6
[[1,3,4,5,6,8],[2,7]]
 => [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[[1,2,4,5,6,8],[3,7]]
 => [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 5
[[1,2,3,5,6,8],[4,7]]
 => [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,6,8],[5,7]]
 => [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,8],[6,7]]
 => [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6
[[1,3,4,5,6,7],[2,8]]
 => [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[[1,2,4,5,6,7],[3,8]]
 => [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[[1,2,3,5,6,7],[4,8]]
 => [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 5
[[1,2,3,4,6,7],[5,8]]
 => [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 5
[[1,2,3,4,5,7],[6,8]]
 => [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6
[[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[[1,3,5,6,7,8],[2],[4]]
 => [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,5,6,7,8],[3],[4]]
 => [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[[1,3,4,6,7,8],[2],[5]]
 => [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,4,6,7,8],[3],[5]]
 => [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,3,6,7,8],[4],[5]]
 => [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,3,4,5,7,8],[2],[6]]
 => [6,2,1,3,4,5,7,8] => ?
 => ?
 => ? = 5
[[1,2,4,5,7,8],[3],[6]]
 => [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,3,5,7,8],[4],[6]]
 => [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,3,4,7,8],[5],[6]]
 => [6,5,1,2,3,4,7,8] => ?
 => ?
 => ? = 5
[[1,3,4,5,6,8],[2],[7]]
 => [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[[1,2,4,5,6,8],[3],[7]]
 => [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 5
[[1,2,3,5,6,8],[4],[7]]
 => [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,6,8],[5],[7]]
 => [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,5,8],[6],[7]]
 => [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 5
[[1,3,4,5,6,7],[2],[8]]
 => [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[[1,2,4,5,6,7],[3],[8]]
 => [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[[1,2,3,5,6,7],[4],[8]]
 => [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 5
[[1,2,3,4,6,7],[5],[8]]
 => [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 5
[[1,2,3,4,5,7],[6],[8]]
 => [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5
[[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5
[[1,3,5,7,8],[2,4,6]]
 => [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 4

Description

The number of valleys of the Dyck path.

The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000245The number of ascents of a permutation. St000632The jump number of the poset. St000167The number of leaves of an ordered tree. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000021The number of descents of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama 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. 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. St001812The biclique partition number of a graph. St000039The number of crossings of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St001330The hat guessing number of a graph. St001726The number of visible inversions of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001896The number of right descents of a signed permutations.