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Your data matches 108 different statistics following compositions of up to 3 maps.
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Matching statistic: St001038
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000297
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[2,2]
=> [2]
=> [1,1]
=> 110 => 2
[2,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[3,2]
=> [2]
=> [1,1]
=> 110 => 2
[3,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[4,2]
=> [2]
=> [1,1]
=> 110 => 2
[4,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[3,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 1100 => 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 10010 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
[5,2]
=> [2]
=> [1,1]
=> 110 => 2
[5,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[4,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 1
[3,2,2]
=> [2,2]
=> [2,2]
=> 1100 => 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 10010 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 10100 => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[6,2]
=> [2]
=> [1,1]
=> 110 => 2
[6,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[5,3]
=> [3]
=> [1,1,1]
=> 1110 => 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 1010 => 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 1000 => 1
[4,4]
=> [4]
=> [1,1,1,1]
=> 11110 => 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 10110 => 1
[4,2,2]
=> [2,2]
=> [2,2]
=> 1100 => 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 10010 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 11010 => 2
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
[3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 10100 => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 11000 => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 100100 => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> 110 => 110 => 1
[2,2]
=> [2]
=> 100 => 010 => 2
[2,1,1]
=> [1,1]
=> 110 => 110 => 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 1
[3,2]
=> [2]
=> 100 => 010 => 2
[3,1,1]
=> [1,1]
=> 110 => 110 => 1
[2,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 1
[4,2]
=> [2]
=> 100 => 010 => 2
[4,1,1]
=> [1,1]
=> 110 => 110 => 1
[3,3]
=> [3]
=> 1000 => 0010 => 3
[3,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 1
[2,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 1
[5,2]
=> [2]
=> 100 => 010 => 2
[5,1,1]
=> [1,1]
=> 110 => 110 => 1
[4,3]
=> [3]
=> 1000 => 0010 => 3
[4,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 1
[3,3,1]
=> [3,1]
=> 10010 => 10100 => 1
[3,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 1
[2,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => 1
[6,2]
=> [2]
=> 100 => 010 => 2
[6,1,1]
=> [1,1]
=> 110 => 110 => 1
[5,3]
=> [3]
=> 1000 => 0010 => 3
[5,2,1]
=> [2,1]
=> 1010 => 1100 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 1
[4,4]
=> [4]
=> 10000 => 00010 => 4
[4,3,1]
=> [3,1]
=> 10010 => 10100 => 1
[4,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => 11010 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 1
[3,3,2]
=> [3,2]
=> 10100 => 01100 => 2
[3,3,1,1]
=> [3,1,1]
=> 100110 => 101010 => 1
[3,2,2,1]
=> [2,2,1]
=> 11010 => 11100 => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 110110 => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 1
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01110 => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 111010 => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [2]
=> 1
[2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,2]
=> [2]
=> [1,1]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[4,2]
=> [2]
=> [1,1]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[5,2]
=> [2]
=> [1,1]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[6,2]
=> [2]
=> [1,1]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[4,4]
=> [4]
=> [1,1,1,1]
=> 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> ? = 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 2
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> ? = 2
[4,4,4,4,1]
=> [4,4,4,1]
=> [4,3,3,3]
=> ? = 1
[4,4,4,3,2]
=> [4,4,3,2]
=> [4,4,3,2]
=> ? = 2
[4,4,3,3,3]
=> [4,3,3,3]
=> [4,4,4,1]
=> ? = 3
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> ? = 1
[4,4,3,2,2,2]
=> [4,3,2,2,2]
=> [5,5,2,1]
=> ? = 2
[4,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> ? = 1
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> ? = 2
[4,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> ? = 2
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 2
[6,4,3,3,2,1]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> ? = 1
[5,4,3,3,2,1]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> ? = 1
[6,5,4,3,2]
=> [5,4,3,2]
=> [4,4,3,2,1]
=> ? = 2
[5,5,4,3,2]
=> [5,4,3,2]
=> [4,4,3,2,1]
=> ? = 2
[6,4,4,3,2]
=> [4,4,3,2]
=> [4,4,3,2]
=> ? = 2
[5,4,4,3,2]
=> [4,4,3,2]
=> [4,4,3,2]
=> ? = 2
[6,5,3,3,2]
=> [5,3,3,2]
=> [4,4,3,1,1]
=> ? = 2
[5,5,3,3,2]
=> [5,3,3,2]
=> [4,4,3,1,1]
=> ? = 2
[6,5,4,2,2]
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 2
[5,5,4,2,2]
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 2
[6,5,4,3,1]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> ? = 1
[5,5,4,3,1]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> ? = 1
[5,4,3,3,3,2]
=> [4,3,3,3,2]
=> [5,5,4,1]
=> ? = 2
[4,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> ? = 2
[5,4,3,2,2,2]
=> [4,3,2,2,2]
=> [5,5,2,1]
=> ? = 2
[7,4,3,3,2,1]
=> [4,3,3,2,1]
=> [5,4,3,1]
=> ? = 1
[5,4,4,4,3]
=> [4,4,4,3]
=> [4,4,4,3]
=> ? = 3
[5,4,3,3,3]
=> [4,3,3,3]
=> [4,4,4,1]
=> ? = 3
[4,4,4,4,2]
=> [4,4,4,2]
=> [4,4,3,3]
=> ? = 2
[7,5,4,3,2]
=> [5,4,3,2]
=> [4,4,3,2,1]
=> ? = 2
[7,5,4,3,1]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> ? = 1
[7,6,5,4]
=> [6,5,4]
=> [3,3,3,3,2,1]
=> ? = 4
[6,5,5,4]
=> [5,5,4]
=> [3,3,3,3,2]
=> ? = 4
[7,6,5,3]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3
[7,6,4,3]
=> [6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3
[7,6,5,2]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> ? = 2
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 3
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> ? = 3
[4,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4,4]
=> ? = 4
[5,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> ? = 5
[8,7,6]
=> [7,6]
=> [2,2,2,2,2,2,1]
=> ? = 6
[9,6,5,3]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3
[9,6,4,3]
=> [6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3
[6,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> ? = 5
[5,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4,4]
=> ? = 4
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000733
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[4,4]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[5,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
[5,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[5,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
[4,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[4,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> [[1,2,3,4,13],[5,6,7,8],[9,10,11,12]]
=> ? = 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 2
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> [[1,3,4,5,6,7],[2,9,10,11,12,13],[8]]
=> ? = 2
[6,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 5
[6,6,4,1]
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 1
[6,6,3,2]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 2
[6,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[6,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
[6,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[6,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[5,5,5,2]
=> [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[5,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4
[5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
[5,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[5,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[5,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[4,4,4,4,1]
=> [4,4,4,1]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 1
[4,4,3,3,3]
=> [4,3,3,3]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 3
[4,4,3,2,2,2]
=> [4,3,2,2,2]
=> [5,5,2,1]
=> [[1,3,6,7,8],[2,5,11,12,13],[4,10],[9]]
=> ? = 2
[4,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> [[1,2,3,4,13],[5,6,7,8],[9,10,11,12]]
=> ? = 1
[4,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 2
[4,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> [[1,3,4,5,6,7],[2,9,10,11,12,13],[8]]
=> ? = 2
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 2
[7,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[5,4,3,3,3,2]
=> [4,3,3,3,2]
=> [5,5,4,1]
=> [[1,3,4,5,10],[2,7,8,9,15],[6,12,13,14],[11]]
=> ? = 2
[4,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 2
[5,4,3,2,2,2]
=> [4,3,2,2,2]
=> [5,5,2,1]
=> [[1,3,6,7,8],[2,5,11,12,13],[4,10],[9]]
=> ? = 2
[5,4,4,4,3]
=> [4,4,4,3]
=> [4,4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10,15],[12,13,14]]
=> ? = 3
[5,4,3,3,3]
=> [4,3,3,3]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 3
[7,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[4,4,4,4,2]
=> [4,4,4,2]
=> [4,4,3,3]
=> [[1,2,3,10],[4,5,6,14],[7,8,9],[11,12,13]]
=> ? = 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000383
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> 110 => [2,1] => 1
[2,2]
=> [2]
=> 100 => [1,2] => 2
[2,1,1]
=> [1,1]
=> 110 => [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => [3,1] => 1
[3,2]
=> [2]
=> 100 => [1,2] => 2
[3,1,1]
=> [1,1]
=> 110 => [2,1] => 1
[2,2,1]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => [3,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => [4,1] => 1
[4,2]
=> [2]
=> 100 => [1,2] => 2
[4,1,1]
=> [1,1]
=> 110 => [2,1] => 1
[3,3]
=> [3]
=> 1000 => [1,3] => 3
[3,2,1]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => [3,1] => 1
[2,2,2]
=> [2,2]
=> 1100 => [2,2] => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => [4,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => [5,1] => 1
[5,2]
=> [2]
=> 100 => [1,2] => 2
[5,1,1]
=> [1,1]
=> 110 => [2,1] => 1
[4,3]
=> [3]
=> 1000 => [1,3] => 3
[4,2,1]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => [3,1] => 1
[3,3,1]
=> [3,1]
=> 10010 => [1,2,1,1] => 1
[3,2,2]
=> [2,2]
=> 1100 => [2,2] => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => [4,1] => 1
[2,2,2,1]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => [5,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => [6,1] => 1
[6,2]
=> [2]
=> 100 => [1,2] => 2
[6,1,1]
=> [1,1]
=> 110 => [2,1] => 1
[5,3]
=> [3]
=> 1000 => [1,3] => 3
[5,2,1]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => [3,1] => 1
[4,4]
=> [4]
=> 10000 => [1,4] => 4
[4,3,1]
=> [3,1]
=> 10010 => [1,2,1,1] => 1
[4,2,2]
=> [2,2]
=> 1100 => [2,2] => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => [4,1] => 1
[3,3,2]
=> [3,2]
=> 10100 => [1,1,1,2] => 2
[3,3,1,1]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[3,2,2,1]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => [5,1] => 1
[2,2,2,2]
=> [2,2,2]
=> 11100 => [3,2] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => [6,1] => 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> 10111100 => [1,1,4,2] => ? = 2
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> 101110110 => [1,1,3,1,2,1] => ? = 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> 101010110 => [1,1,1,1,1,1,2,1] => ? = 1
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> 100111010 => [1,2,3,1,1,1] => ? = 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> 10111100 => [1,1,4,2] => ? = 2
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> 101110110 => [1,1,3,1,2,1] => ? = 1
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> 11101010 => [3,1,1,1,1,1] => ? = 1
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[3,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[5,5,5,1]
=> [5,5,1]
=> 11000010 => [2,4,1,1] => ? = 1
[5,5,4,1,1]
=> [5,4,1,1]
=> 101000110 => [1,1,1,3,2,1] => ? = 1
[5,5,3,2,1]
=> [5,3,2,1]
=> 100101010 => [1,2,1,1,1,1,1,1] => ? = 1
[5,5,2,2,2]
=> [5,2,2,2]
=> 100011100 => [1,3,3,2] => ? = 2
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> 101010110 => [1,1,1,1,1,1,2,1] => ? = 1
[5,4,2,2,2,1]
=> [4,2,2,2,1]
=> 100111010 => [1,2,3,1,1,1] => ? = 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> 10111100 => [1,1,4,2] => ? = 2
[5,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> 101110110 => [1,1,3,1,2,1] => ? = 1
[4,4,3,3,1,1]
=> [4,3,3,1,1]
=> 101100110 => [1,1,2,2,2,1] => ? = 1
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> 101011010 => [1,1,1,1,2,1,1,1] => ? = 1
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> 100111100 => [1,2,4,2] => ? = 2
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> 11101010 => [3,1,1,1,1,1] => ? = 1
[4,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[4,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> 11110010 => [4,2,1,1] => ? = 1
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> 101111100 => [1,1,5,2] => ? = 2
[6,6,5]
=> [6,5]
=> 10100000 => [1,1,1,5] => ? = 5
[6,6,4,1]
=> [6,4,1]
=> 100100010 => [1,2,1,3,1,1] => ? = 1
[6,6,3,2]
=> [6,3,2]
=> 100010100 => [1,3,1,1,1,2] => ? = 2
[6,5,5,1]
=> [5,5,1]
=> 11000010 => [2,4,1,1] => ? = 1
[6,5,4,1,1]
=> [5,4,1,1]
=> 101000110 => [1,1,1,3,2,1] => ? = 1
[6,5,3,2,1]
=> [5,3,2,1]
=> 100101010 => [1,2,1,1,1,1,1,1] => ? = 1
[6,5,2,2,2]
=> [5,2,2,2]
=> 100011100 => [1,3,3,2] => ? = 2
[6,4,3,2,1,1]
=> [4,3,2,1,1]
=> 101010110 => [1,1,1,1,1,1,2,1] => ? = 1
[6,4,2,2,2,1]
=> [4,2,2,2,1]
=> 100111010 => [1,2,3,1,1,1] => ? = 1
[6,3,2,2,2,2]
=> [3,2,2,2,2]
=> 10111100 => [1,1,4,2] => ? = 2
[6,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> 101110110 => [1,1,3,1,2,1] => ? = 1
[5,5,4,3]
=> [5,4,3]
=> 10101000 => [1,1,1,1,1,3] => ? = 3
[5,5,4,2,1]
=> [5,4,2,1]
=> 101001010 => [1,1,1,2,1,1,1,1] => ? = 1
[5,5,3,3,1]
=> [5,3,3,1]
=> 100110010 => [1,2,2,2,1,1] => ? = 1
[5,5,3,2,2]
=> [5,3,2,2]
=> 100101100 => [1,2,1,1,2,2] => ? = 2
[5,4,3,3,1,1]
=> [4,3,3,1,1]
=> 101100110 => [1,1,2,2,2,1] => ? = 1
[5,4,3,2,2,1]
=> [4,3,2,2,1]
=> 101011010 => [1,1,1,1,2,1,1,1] => ? = 1
[5,4,2,2,2,2]
=> [4,2,2,2,2]
=> 100111100 => [1,2,4,2] => ? = 2
[5,3,3,3,2,1]
=> [3,3,3,2,1]
=> 11101010 => [3,1,1,1,1,1] => ? = 1
[5,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[5,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[4,4,3,3,2,1]
=> [4,3,3,2,1]
=> 101101010 => [1,1,2,1,1,1,1,1] => ? = 1
[4,4,3,2,2,2]
=> [4,3,2,2,2]
=> 101011100 => [1,1,1,1,3,2] => ? = 2
[4,3,3,3,3,1]
=> [3,3,3,3,1]
=> 11110010 => [4,2,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[3,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[4,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [3] => 3
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[5,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [3] => 3
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[6,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [3] => 3
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,3] => ? = 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,3] => ? = 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 2
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [1,2,4,4] => ? = 1
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [1,3,3,4] => ? = 1
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [2,2,3,4] => ? = 2
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [1,1,2,3,4] => ? = 1
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [1,2,2,2,4] => ? = 1
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,3] => ? = 1
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,3] => ? = 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 2
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 2
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[5,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 1
[5,5,4,2]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [2,4,5] => ? = 2
[5,5,4,1,1]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [1,1,4,5] => ? = 1
[5,5,3,3]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [3,3,5] => ? = 3
[5,5,3,2,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [1,2,3,5] => ? = 1
[5,5,2,2,2]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [2,2,2,5] => ? = 2
[5,4,4,2,1]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [1,2,4,4] => ? = 1
[5,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [1,3,3,4] => ? = 1
[5,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [2,2,3,4] => ? = 2
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [1,1,2,3,4] => ? = 1
[5,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [1,2,2,2,4] => ? = 1
[5,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,3] => ? = 1
[5,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,3] => ? = 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 2
[5,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[5,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[4,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 4
[4,4,3,3,1,1]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,4] => ? = 1
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 2
[4,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 2
[4,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? => ? = 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? = 2
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 2
[6,6,5]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 5
[6,6,4,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 1
[6,6,3,2]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 2
[6,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 1
[6,5,4,2]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [2,4,5] => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000745
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[2,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[3,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[4,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[5,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[6,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 2
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 2
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 3
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 1
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 1
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 2
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 1
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> ? = 1
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 2
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 1
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 2
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 3
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 2
[3,3,2,2,2,2,1]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[5,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
=> ? = 1
[5,5,4,2]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 2
[5,5,4,1,1]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 1
[5,5,3,3]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> ? = 3
[5,5,3,2,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 1
[5,5,2,2,2]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 2
[5,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 3
[5,4,4,2,1]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 1
[5,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 1
[5,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 2
[5,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 1
[5,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> ? = 1
[5,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 2
[5,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 1
[5,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 1
[5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 2
[5,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
[5,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[4,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
=> ? = 4
[4,4,4,3,1]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
=> ? = 1
[4,4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
[4,4,3,3,2]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 2
[4,4,3,3,1,1]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11],[12]]
=> ? = 1
[4,4,3,2,2,1]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
=> ? = 1
[4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 2
[4,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 3
[4,3,3,3,2,1]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000990
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 78%●distinct values known / distinct values provided: 62%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 78%●distinct values known / distinct values provided: 62%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 1
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 6
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 1
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ? = 2
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 6
[7,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[6,6,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 1
[6,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[6,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[5,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[5,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 1
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 1
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 1
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ? = 2
[8,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 6
[8,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[7,7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ? = 7
[7,6,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 1
[7,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[7,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[6,6,2]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 2
[6,6,1,1]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 1
[6,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[6,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
Matching statistic: St000657
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[3,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[4,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [3] => 3
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[5,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [3] => 3
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[6,2]
=> [2]
=> [[1,2]]
=> [2] => 2
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [3] => 3
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 1
[3,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 1
[3,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 1
[4,4,4,2]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[4,4,4,1,1]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => ? = 1
[4,4,3,3]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 3
[4,4,3,2,1]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[4,4,3,1,1,1]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 1
[4,4,2,2,2]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[4,4,2,2,1,1]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 1
[4,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 1
[4,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[4,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[4,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[4,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[4,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[4,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 2
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? = 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? = 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 2
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? = 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [2,2,2,2,2,2] => ? = 2
[5,5,5]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 5
[5,5,4,1]
=> [5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [5,4,1] => ? = 1
[5,5,3,2]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [5,3,2] => ? = 2
[5,5,3,1,1]
=> [5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [5,3,1,1] => ? = 1
[5,5,2,2,1]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [5,2,2,1] => ? = 1
[5,4,4,2]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[5,4,4,1,1]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => ? = 1
[5,4,3,3]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 3
[5,4,3,2,1]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[5,4,3,1,1,1]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 1
[5,4,2,2,2]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[5,4,2,2,1,1]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 1
[5,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 1
[5,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[5,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[5,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[5,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[5,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[5,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 1
Description
The smallest part of an integer composition.
The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000974The length of the trunk of an ordered tree. St001075The minimal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000654The first descent of a permutation. St000090The variation of a composition. St000314The number of left-to-right-maxima of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000011The number of touch points (or returns) of a Dyck path. St000439The position of the first down step of a Dyck path. St000054The first entry of the permutation. St000617The number of global maxima of a Dyck path. St000674The number of hills of a Dyck path. St000700The protection number of an ordered tree. St001050The number of terminal closers of a set partition. St000234The number of global ascents of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000221The number of strong fixed points of a permutation. St000989The number of final rises of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000007The number of saliances of the permutation. St000883The number of longest increasing subsequences of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000546The number of global descents of a permutation. St000025The number of initial rises of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000052The number of valleys of a Dyck path not on the x-axis. St000068The number of minimal elements in a poset. St000765The number of weak records in an integer composition. St000908The length of the shortest maximal antichain in a poset. St001810The number of fixed points of a permutation smaller than its largest moved point. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000918The 2-limited packing number of a graph. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000729The minimal arc length of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000565The major index of a set partition. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000991The number of right-to-left minima of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St000374The number of exclusive right-to-left minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000053The number of valleys of the Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000031The number of cycles in the cycle decomposition of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000352The Elizalde-Pak rank of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000260The radius of a connected graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000906The length of the shortest maximal chain in a poset. St000287The number of connected components of a graph. St000258The burning number of a graph. St000315The number of isolated vertices of a graph. St001330The hat guessing number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph.
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