- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000492
(load all 68 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000492: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => {{1},{2}}
 => 1
[2,1] => {{1,2}}
 => 0
[1,2,3] => {{1},{2},{3}}
 => 3
[1,3,2] => {{1},{2,3}}
 => 1
[2,1,3] => {{1,2},{3}}
 => 2
[2,3,1] => {{1,2,3}}
 => 0
[3,1,2] => {{1,3},{2}}
 => 1
[3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => 6
[1,2,4,3] => {{1},{2},{3,4}}
 => 3
[1,3,2,4] => {{1},{2,3},{4}}
 => 4
[1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,4,2,3] => {{1},{2,4},{3}}
 => 3
[1,4,3,2] => {{1},{2,4},{3}}
 => 3
[2,1,3,4] => {{1,2},{3},{4}}
 => 5
[2,1,4,3] => {{1,2},{3,4}}
 => 2
[2,3,1,4] => {{1,2,3},{4}}
 => 3
[2,3,4,1] => {{1,2,3,4}}
 => 0
[2,4,1,3] => {{1,2,4},{3}}
 => 2
[2,4,3,1] => {{1,2,4},{3}}
 => 2
[3,1,2,4] => {{1,3},{2},{4}}
 => 4
[3,1,4,2] => {{1,3,4},{2}}
 => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => 4
[3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => 1
[3,4,2,1] => {{1,3},{2,4}}
 => 1
[4,1,2,3] => {{1,4},{2},{3}}
 => 3
[4,1,3,2] => {{1,4},{2},{3}}
 => 3
[4,2,1,3] => {{1,4},{2},{3}}
 => 3
[4,2,3,1] => {{1,4},{2},{3}}
 => 3
[4,3,1,2] => {{1,4},{2,3}}
 => 1
[4,3,2,1] => {{1,4},{2,3}}
 => 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 10
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 6
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 7
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 6
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 6
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 8
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 4
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 5
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
 => 4
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 4
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => 7
[1,4,2,5,3] => {{1},{2,4,5},{3}}
 => 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 7
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 3
[1,4,5,3,2] => {{1},{2,4},{3,5}}
 => 3

Description

The rob statistic of a set partition. Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ . According to [1, Definition 3], a '''rob''' (right-opener-bigger) of

$S$ is given by a pair

$i < j$ such that

$j = \operatorname{min} B_b$ and

$i \in B_a$ for

$a < b$ . This is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.

Matching statistic: St000391

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => 0 => 1 => 1
[2,1] => [2,1] => 1 => 0 => 0
[1,2,3] => [1,2,3] => 00 => 11 => 3
[1,3,2] => [1,3,2] => 01 => 10 => 1
[2,1,3] => [2,1,3] => 10 => 01 => 2
[2,3,1] => [3,2,1] => 11 => 00 => 0
[3,1,2] => [3,1,2] => 01 => 10 => 1
[3,2,1] => [2,3,1] => 01 => 10 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 111 => 6
[1,2,4,3] => [1,2,4,3] => 001 => 110 => 3
[1,3,2,4] => [1,3,2,4] => 010 => 101 => 4
[1,3,4,2] => [1,4,3,2] => 011 => 100 => 1
[1,4,2,3] => [1,4,2,3] => 001 => 110 => 3
[1,4,3,2] => [1,3,4,2] => 001 => 110 => 3
[2,1,3,4] => [2,1,3,4] => 100 => 011 => 5
[2,1,4,3] => [2,1,4,3] => 101 => 010 => 2
[2,3,1,4] => [3,2,1,4] => 110 => 001 => 3
[2,3,4,1] => [4,3,2,1] => 111 => 000 => 0
[2,4,1,3] => [4,2,1,3] => 101 => 010 => 2
[2,4,3,1] => [3,4,2,1] => 101 => 010 => 2
[3,1,2,4] => [3,1,2,4] => 010 => 101 => 4
[3,1,4,2] => [4,3,1,2] => 011 => 100 => 1
[3,2,1,4] => [2,3,1,4] => 010 => 101 => 4
[3,2,4,1] => [2,4,3,1] => 011 => 100 => 1
[3,4,1,2] => [4,1,3,2] => 011 => 100 => 1
[3,4,2,1] => [4,2,3,1] => 011 => 100 => 1
[4,1,2,3] => [4,1,2,3] => 001 => 110 => 3
[4,1,3,2] => [3,4,1,2] => 001 => 110 => 3
[4,2,1,3] => [2,4,1,3] => 001 => 110 => 3
[4,2,3,1] => [2,3,4,1] => 001 => 110 => 3
[4,3,1,2] => [3,1,4,2] => 011 => 100 => 1
[4,3,2,1] => [3,2,4,1] => 011 => 100 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 1111 => 10
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1110 => 6
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1101 => 7
[1,2,4,5,3] => [1,2,5,4,3] => 0011 => 1100 => 3
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1110 => 6
[1,2,5,4,3] => [1,2,4,5,3] => 0001 => 1110 => 6
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1011 => 8
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 4
[1,3,4,2,5] => [1,4,3,2,5] => 0110 => 1001 => 5
[1,3,4,5,2] => [1,5,4,3,2] => 0111 => 1000 => 1
[1,3,5,2,4] => [1,5,3,2,4] => 0101 => 1010 => 4
[1,3,5,4,2] => [1,4,5,3,2] => 0101 => 1010 => 4
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 1101 => 7
[1,4,2,5,3] => [1,5,4,2,3] => 0011 => 1100 => 3
[1,4,3,2,5] => [1,3,4,2,5] => 0010 => 1101 => 7
[1,4,3,5,2] => [1,3,5,4,2] => 0011 => 1100 => 3
[1,4,5,2,3] => [1,5,2,4,3] => 0011 => 1100 => 3
[1,4,5,3,2] => [1,5,3,4,2] => 0011 => 1100 => 3
[2,4,5,6,1,7,8,3] => [8,7,6,2,1,5,4,3] => ? => ? => ? = 2
[3,6,4,5,7,2,8,1] => [8,7,5,4,2,6,3,1] => ? => ? => ? = 1
[3,2,5,1,6,7,8,4] => [2,8,7,6,5,3,1,4] => ? => ? => ? = 4

Description

The sum of the positions of the ones in a binary word.

Matching statistic: St000472
(load all 3 compositions to match this statistic)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 95%

Values

[1,2] => [2,1] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [2,3,1] => [3,2,1] => [1,2,3] => 3
[1,3,2] => [3,2,1] => [2,3,1] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[1,2,3,4] => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 6
[1,2,4,3] => [2,4,3,1] => [3,4,2,1] => [1,2,4,3] => 3
[1,3,2,4] => [3,2,4,1] => [2,4,3,1] => [1,3,4,2] => 4
[1,3,4,2] => [4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 1
[1,4,2,3] => [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 3
[1,4,3,2] => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 3
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 5
[2,1,4,3] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,2,4] => [3,1,4,2] => [4,3,1,2] => [2,1,3,4] => 4
[3,1,4,2] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 4
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 1
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 1
[4,1,2,3] => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 3
[4,1,3,2] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 3
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [3,1,2,4] => 3
[4,2,3,1] => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[4,3,1,2] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[4,3,2,1] => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 10
[1,2,3,5,4] => [2,3,5,4,1] => [4,5,3,2,1] => [1,2,3,5,4] => 6
[1,2,4,3,5] => [2,4,3,5,1] => [3,5,4,2,1] => [1,2,4,5,3] => 7
[1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,2,1] => [1,2,5,4,3] => 3
[1,2,5,3,4] => [2,4,5,3,1] => [5,3,4,2,1] => [1,2,4,3,5] => 6
[1,2,5,4,3] => [2,5,4,3,1] => [4,3,5,2,1] => [1,2,5,3,4] => 6
[1,3,2,4,5] => [3,2,4,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 8
[1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,3,1] => [1,3,5,4,2] => 4
[1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,4,1] => [1,4,5,3,2] => 5
[1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [2,5,3,4,1] => [1,4,3,5,2] => 4
[1,3,5,4,2] => [5,2,4,3,1] => [2,4,3,5,1] => [1,5,3,4,2] => 4
[1,4,2,3,5] => [3,4,2,5,1] => [5,4,2,3,1] => [1,3,2,4,5] => 7
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,2,3,1] => [1,3,2,5,4] => 3
[1,4,3,2,5] => [4,3,2,5,1] => [3,2,5,4,1] => [1,4,5,2,3] => 7
[1,4,3,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => [1,5,4,2,3] => 3
[1,4,5,2,3] => [4,5,2,3,1] => [5,2,3,4,1] => [1,4,3,2,5] => 3
[1,4,5,3,2] => [5,4,2,3,1] => [4,2,3,5,1] => [1,5,3,2,4] => 3
[1,3,4,2,6,5,7] => [4,2,3,6,5,7,1] => [2,3,5,7,6,4,1] => [1,4,6,7,5,3,2] => ? = 11
[1,3,4,2,6,7,5] => [4,2,3,7,5,6,1] => [2,3,5,6,7,4,1] => [1,4,7,6,5,3,2] => ? = 5
[1,3,4,2,7,5,6] => [4,2,3,6,7,5,1] => [2,3,7,5,6,4,1] => [1,4,6,5,7,3,2] => ? = 10
[1,3,4,2,7,6,5] => [4,2,3,7,6,5,1] => [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 10
[1,3,4,5,2,6,7] => [5,2,3,4,6,7,1] => [2,3,4,7,6,5,1] => [1,5,6,7,4,3,2] => ? = 12
[1,3,4,5,2,7,6] => [5,2,3,4,7,6,1] => [2,3,4,6,7,5,1] => [1,5,7,6,4,3,2] => ? = 6
[1,3,4,5,6,2,7] => [6,2,3,4,5,7,1] => [2,3,4,5,7,6,1] => [1,6,7,5,4,3,2] => ? = 7
[1,3,4,5,6,7,2] => [7,2,3,4,5,6,1] => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 1
[1,3,4,5,7,2,6] => [6,2,3,4,7,5,1] => [2,3,4,7,5,6,1] => [1,6,5,7,4,3,2] => ? = 6
[1,3,4,5,7,6,2] => [7,2,3,4,6,5,1] => [2,3,4,6,5,7,1] => [1,7,5,6,4,3,2] => ? = 6
[1,3,4,6,2,5,7] => [5,2,3,6,4,7,1] => [2,3,7,6,4,5,1] => [1,5,4,6,7,3,2] => ? = 11
[1,3,4,6,2,7,5] => [5,2,3,7,4,6,1] => [2,3,6,7,4,5,1] => [1,5,4,7,6,3,2] => ? = 5
[1,3,4,6,5,2,7] => [6,2,3,5,4,7,1] => [2,3,5,4,7,6,1] => [1,6,7,4,5,3,2] => ? = 11
[1,3,4,6,5,7,2] => [7,2,3,5,4,6,1] => [2,3,5,4,6,7,1] => [1,7,6,4,5,3,2] => ? = 5
[1,3,4,6,7,2,5] => [6,2,3,7,4,5,1] => [2,3,7,4,5,6,1] => [1,6,5,4,7,3,2] => ? = 5
[1,3,4,6,7,5,2] => [7,2,3,6,4,5,1] => [2,3,6,4,5,7,1] => [1,7,5,4,6,3,2] => ? = 5
[1,3,4,7,2,5,6] => [5,2,3,6,7,4,1] => [2,3,7,4,6,5,1] => [1,5,6,4,7,3,2] => ? = 10
[1,3,4,7,2,6,5] => [5,2,3,7,6,4,1] => [2,3,6,4,7,5,1] => [1,5,7,4,6,3,2] => ? = 10
[1,3,4,7,5,2,6] => [6,2,3,5,7,4,1] => [2,3,7,5,4,6,1] => [1,6,4,5,7,3,2] => ? = 10
[1,3,4,7,5,6,2] => [7,2,3,5,6,4,1] => [2,3,6,5,4,7,1] => [1,7,4,5,6,3,2] => ? = 10
[1,3,4,7,6,2,5] => [6,2,3,7,5,4,1] => [2,3,5,7,4,6,1] => [1,6,4,7,5,3,2] => ? = 5
[1,3,4,7,6,5,2] => [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => [1,7,4,6,5,3,2] => ? = 5
[1,3,5,4,2,6,7] => [5,2,4,3,6,7,1] => [2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => ? = 15
[1,3,5,4,2,7,6] => [5,2,4,3,7,6,1] => [2,4,3,6,7,5,1] => [1,5,7,6,3,4,2] => ? = 9
[1,3,5,4,6,2,7] => [6,2,4,3,5,7,1] => [2,4,3,5,7,6,1] => [1,6,7,5,3,4,2] => ? = 10
[1,3,5,4,6,7,2] => [7,2,4,3,5,6,1] => [2,4,3,5,6,7,1] => [1,7,6,5,3,4,2] => ? = 4
[1,3,5,4,7,2,6] => [6,2,4,3,7,5,1] => [2,4,3,7,5,6,1] => [1,6,5,7,3,4,2] => ? = 9
[1,3,5,4,7,6,2] => [7,2,4,3,6,5,1] => [2,4,3,6,5,7,1] => [1,7,5,6,3,4,2] => ? = 9
[1,3,5,6,2,4,7] => [5,2,6,3,4,7,1] => [2,7,6,3,4,5,1] => [1,5,4,3,6,7,2] => ? = 10
[1,3,5,6,2,7,4] => [5,2,7,3,4,6,1] => [2,6,7,3,4,5,1] => [1,5,4,3,7,6,2] => ? = 4
[1,3,5,6,4,2,7] => [6,2,5,3,4,7,1] => [2,5,3,4,7,6,1] => [1,6,7,4,3,5,2] => ? = 10
[1,3,5,6,4,7,2] => [7,2,5,3,4,6,1] => [2,5,3,4,6,7,1] => [1,7,6,4,3,5,2] => ? = 4
[1,3,5,6,7,2,4] => [6,2,7,3,4,5,1] => [2,7,3,4,5,6,1] => [1,6,5,4,3,7,2] => ? = 4
[1,3,5,6,7,4,2] => [7,2,6,3,4,5,1] => [2,6,3,4,5,7,1] => [1,7,5,4,3,6,2] => ? = 4
[1,3,5,7,2,4,6] => [5,2,6,3,7,4,1] => [2,7,3,4,6,5,1] => [1,5,6,4,3,7,2] => ? = 9
[1,3,5,7,2,6,4] => [5,2,7,3,6,4,1] => [2,6,3,4,7,5,1] => [1,5,7,4,3,6,2] => ? = 9
[1,3,5,7,4,2,6] => [6,2,5,3,7,4,1] => [2,7,5,3,4,6,1] => [1,6,4,3,5,7,2] => ? = 9
[1,3,5,7,4,6,2] => [7,2,5,3,6,4,1] => [2,6,5,3,4,7,1] => [1,7,4,3,5,6,2] => ? = 9
[1,3,5,7,6,2,4] => [6,2,7,3,5,4,1] => [2,5,7,3,4,6,1] => [1,6,4,3,7,5,2] => ? = 4
[1,3,5,7,6,4,2] => [7,2,6,3,5,4,1] => [2,5,6,3,4,7,1] => [1,7,4,3,6,5,2] => ? = 4
[1,3,6,2,5,4,7] => [4,2,6,5,3,7,1] => [2,5,3,7,6,4,1] => [1,4,6,7,3,5,2] => ? = 14
[1,3,6,2,5,7,4] => [4,2,7,5,3,6,1] => [2,5,3,6,7,4,1] => [1,4,7,6,3,5,2] => ? = 8
[1,3,6,2,7,5,4] => [4,2,7,6,3,5,1] => [2,6,3,5,7,4,1] => [1,4,7,5,3,6,2] => ? = 8
[1,3,6,4,2,5,7] => [5,2,4,6,3,7,1] => [2,7,6,4,3,5,1] => [1,5,3,4,6,7,2] => ? = 14
[1,3,6,4,2,7,5] => [5,2,4,7,3,6,1] => [2,6,7,4,3,5,1] => [1,5,3,4,7,6,2] => ? = 8
[1,3,6,4,5,2,7] => [6,2,4,5,3,7,1] => [2,5,4,3,7,6,1] => [1,6,7,3,4,5,2] => ? = 14
[1,3,6,4,5,7,2] => [7,2,4,5,3,6,1] => [2,5,4,3,6,7,1] => [1,7,6,3,4,5,2] => ? = 8
[1,3,6,4,7,2,5] => [6,2,4,7,3,5,1] => [2,7,4,3,5,6,1] => [1,6,5,3,4,7,2] => ? = 8
[1,3,6,4,7,5,2] => [7,2,4,6,3,5,1] => [2,6,4,3,5,7,1] => [1,7,5,3,4,6,2] => ? = 8
[1,3,6,5,2,4,7] => [5,2,6,4,3,7,1] => [2,4,7,6,3,5,1] => [1,5,3,6,7,4,2] => ? = 10

Description

The sum of the ascent bottoms of a permutation.

Matching statistic: St000154
(load all 3 compositions to match this statistic)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 73%

Values

[1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [2,3,1] => [3,2,1] => 3
[1,3,2] => [3,2,1] => [2,3,1] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 2
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 6
[1,2,4,3] => [2,4,3,1] => [3,4,2,1] => 3
[1,3,2,4] => [3,2,4,1] => [2,4,3,1] => 4
[1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 1
[1,4,2,3] => [3,4,2,1] => [4,2,3,1] => 3
[1,4,3,2] => [4,3,2,1] => [3,2,4,1] => 3
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 5
[2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [4,3,1,2] => 4
[3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 4
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 1
[4,1,2,3] => [3,4,1,2] => [4,1,3,2] => 3
[4,1,3,2] => [4,3,1,2] => [3,1,4,2] => 3
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [2,3,1,4] => [3,2,1,4] => 3
[4,3,1,2] => [4,2,1,3] => [2,4,1,3] => 1
[4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 10
[1,2,3,5,4] => [2,3,5,4,1] => [4,5,3,2,1] => 6
[1,2,4,3,5] => [2,4,3,5,1] => [3,5,4,2,1] => 7
[1,2,4,5,3] => [2,5,3,4,1] => [3,4,5,2,1] => 3
[1,2,5,3,4] => [2,4,5,3,1] => [5,3,4,2,1] => 6
[1,2,5,4,3] => [2,5,4,3,1] => [4,3,5,2,1] => 6
[1,3,2,4,5] => [3,2,4,5,1] => [2,5,4,3,1] => 8
[1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,3,1] => 4
[1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,4,1] => 5
[1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [2,5,3,4,1] => 4
[1,3,5,4,2] => [5,2,4,3,1] => [2,4,3,5,1] => 4
[1,4,2,3,5] => [3,4,2,5,1] => [5,4,2,3,1] => 7
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,2,3,1] => 3
[1,4,3,2,5] => [4,3,2,5,1] => [3,2,5,4,1] => 7
[1,4,3,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => 3
[1,4,5,2,3] => [4,5,2,3,1] => [5,2,3,4,1] => 3
[1,4,5,3,2] => [5,4,2,3,1] => [4,2,3,5,1] => 3
[1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => ? = 21
[1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [6,7,5,4,3,2,1] => ? = 15
[1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [5,7,6,4,3,2,1] => ? = 16
[1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [5,6,7,4,3,2,1] => ? = 10
[1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [7,5,6,4,3,2,1] => ? = 15
[1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [6,5,7,4,3,2,1] => ? = 15
[1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => [4,7,6,5,3,2,1] => ? = 17
[1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => [4,6,7,5,3,2,1] => ? = 11
[1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => [4,5,7,6,3,2,1] => ? = 12
[1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => [4,5,6,7,3,2,1] => ? = 6
[1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => [4,7,5,6,3,2,1] => ? = 11
[1,2,3,5,7,6,4] => [2,3,7,4,6,5,1] => [4,6,5,7,3,2,1] => ? = 11
[1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => [7,6,4,5,3,2,1] => ? = 16
[1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [6,7,4,5,3,2,1] => ? = 10
[1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => [5,4,7,6,3,2,1] => ? = 16
[1,2,3,6,5,7,4] => [2,3,7,5,4,6,1] => [5,4,6,7,3,2,1] => ? = 10
[1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [7,4,5,6,3,2,1] => ? = 10
[1,2,3,6,7,5,4] => [2,3,7,6,4,5,1] => [6,4,5,7,3,2,1] => ? = 10
[1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,4,6,5,3,2,1] => ? = 15
[1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [6,4,7,5,3,2,1] => ? = 15
[1,2,3,7,5,4,6] => [2,3,6,5,7,4,1] => [7,5,4,6,3,2,1] => ? = 15
[1,2,3,7,5,6,4] => [2,3,7,5,6,4,1] => [6,5,4,7,3,2,1] => ? = 15
[1,2,3,7,6,4,5] => [2,3,6,7,5,4,1] => [5,7,4,6,3,2,1] => ? = 10
[1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => [5,6,4,7,3,2,1] => ? = 10
[1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => [3,7,6,5,4,2,1] => ? = 18
[1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => [3,6,7,5,4,2,1] => ? = 12
[1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => [3,5,7,6,4,2,1] => ? = 13
[1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => [3,5,6,7,4,2,1] => ? = 7
[1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => [3,7,5,6,4,2,1] => ? = 12
[1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => [3,6,5,7,4,2,1] => ? = 12
[1,2,4,5,3,6,7] => [2,5,3,4,6,7,1] => [3,4,7,6,5,2,1] => ? = 14
[1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => [3,4,6,7,5,2,1] => ? = 8
[1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => [3,4,5,7,6,2,1] => ? = 9
[1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => [3,4,5,6,7,2,1] => ? = 3
[1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => [3,4,7,5,6,2,1] => ? = 8
[1,2,4,5,7,6,3] => [2,7,3,4,6,5,1] => [3,4,6,5,7,2,1] => ? = 8
[1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => [3,7,6,4,5,2,1] => ? = 13
[1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => [3,6,7,4,5,2,1] => ? = 7
[1,2,4,6,5,3,7] => [2,6,3,5,4,7,1] => [3,5,4,7,6,2,1] => ? = 13
[1,2,4,6,5,7,3] => [2,7,3,5,4,6,1] => [3,5,4,6,7,2,1] => ? = 7
[1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => [3,7,4,5,6,2,1] => ? = 7
[1,2,4,6,7,5,3] => [2,7,3,6,4,5,1] => [3,6,4,5,7,2,1] => ? = 7
[1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [3,7,4,6,5,2,1] => ? = 12
[1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => [3,6,4,7,5,2,1] => ? = 12
[1,2,4,7,5,3,6] => [2,6,3,5,7,4,1] => [3,7,5,4,6,2,1] => ? = 12
[1,2,4,7,5,6,3] => [2,7,3,5,6,4,1] => [3,6,5,4,7,2,1] => ? = 12
[1,2,4,7,6,3,5] => [2,6,3,7,5,4,1] => [3,5,7,4,6,2,1] => ? = 7
[1,2,4,7,6,5,3] => [2,7,3,6,5,4,1] => [3,5,6,4,7,2,1] => ? = 7
[1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => [7,6,5,3,4,2,1] => ? = 17
[1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => [6,7,5,3,4,2,1] => ? = 11

Description

The sum of the descent bottoms of a permutation. This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$ For the descent tops, see [[St000111]].