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Matching statistic: St000989
(load all 62 compositions to match this statistic)
(load all 62 compositions to match this statistic)
St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 0
[1,2,3] => 2
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 0
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 2
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 2
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 1
[3,4,2,1] => 0
[4,1,2,3] => 2
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 0
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 2
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 2
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 0
[1,4,5,2,3] => 1
[1,4,5,3,2] => 0
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St000654
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 3 = 2 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 1 = 0 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 1 = 0 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 3 = 2 + 1
[1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 1 = 0 + 1
[2,1,4,3] => 1 = 0 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 3 = 2 + 1
[2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => 1 = 0 + 1
[3,1,4,2] => 1 = 0 + 1
[3,2,1,4] => 1 = 0 + 1
[3,2,4,1] => 1 = 0 + 1
[3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => 1 = 0 + 1
[4,2,1,3] => 1 = 0 + 1
[4,2,3,1] => 1 = 0 + 1
[4,3,1,2] => 1 = 0 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => 3 = 2 + 1
[1,2,4,5,3] => 4 = 3 + 1
[1,2,5,3,4] => 3 = 2 + 1
[1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => 3 = 2 + 1
[1,3,4,5,2] => 4 = 3 + 1
[1,3,5,2,4] => 3 = 2 + 1
[1,3,5,4,2] => 3 = 2 + 1
[1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => 2 = 1 + 1
[1,4,3,2,5] => 2 = 1 + 1
[1,4,3,5,2] => 2 = 1 + 1
[1,4,5,2,3] => 3 = 2 + 1
[1,4,5,3,2] => 3 = 2 + 1
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000990
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 2 = 1 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 1 = 0 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 1 = 0 + 1
[3,1,2] => 2 = 1 + 1
[3,2,1] => 3 = 2 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 1 = 0 + 1
[1,3,2,4] => 1 = 0 + 1
[1,3,4,2] => 1 = 0 + 1
[1,4,2,3] => 1 = 0 + 1
[1,4,3,2] => 1 = 0 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 1 = 0 + 1
[2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => 1 = 0 + 1
[2,4,3,1] => 1 = 0 + 1
[3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => 1 = 0 + 1
[3,4,2,1] => 1 = 0 + 1
[4,1,2,3] => 2 = 1 + 1
[4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 3 = 2 + 1
[4,3,2,1] => 4 = 3 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 1 = 0 + 1
[1,2,4,3,5] => 1 = 0 + 1
[1,2,4,5,3] => 1 = 0 + 1
[1,2,5,3,4] => 1 = 0 + 1
[1,2,5,4,3] => 1 = 0 + 1
[1,3,2,4,5] => 1 = 0 + 1
[1,3,2,5,4] => 1 = 0 + 1
[1,3,4,2,5] => 1 = 0 + 1
[1,3,4,5,2] => 1 = 0 + 1
[1,3,5,2,4] => 1 = 0 + 1
[1,3,5,4,2] => 1 = 0 + 1
[1,4,2,3,5] => 1 = 0 + 1
[1,4,2,5,3] => 1 = 0 + 1
[1,4,3,2,5] => 1 = 0 + 1
[1,4,3,5,2] => 1 = 0 + 1
[1,4,5,2,3] => 1 = 0 + 1
[1,4,5,3,2] => 1 = 0 + 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: St000297
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 1
[1,2,3] => 00 => 0
[1,3,2] => 01 => 0
[2,1,3] => 10 => 1
[2,3,1] => 01 => 0
[3,1,2] => 10 => 1
[3,2,1] => 11 => 2
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 0
[1,3,2,4] => 010 => 0
[1,3,4,2] => 001 => 0
[1,4,2,3] => 010 => 0
[1,4,3,2] => 011 => 0
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 0
[2,3,4,1] => 001 => 0
[2,4,1,3] => 010 => 0
[2,4,3,1] => 011 => 0
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 2
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 0
[3,4,2,1] => 011 => 0
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 2
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 2
[4,3,2,1] => 111 => 3
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 0
[1,2,4,3,5] => 0010 => 0
[1,2,4,5,3] => 0001 => 0
[1,2,5,3,4] => 0010 => 0
[1,2,5,4,3] => 0011 => 0
[1,3,2,4,5] => 0100 => 0
[1,3,2,5,4] => 0101 => 0
[1,3,4,2,5] => 0010 => 0
[1,3,4,5,2] => 0001 => 0
[1,3,5,2,4] => 0010 => 0
[1,3,5,4,2] => 0011 => 0
[1,4,2,3,5] => 0100 => 0
[1,4,2,5,3] => 0101 => 0
[1,4,3,2,5] => 0110 => 0
[1,4,3,5,2] => 0101 => 0
[1,4,5,2,3] => 0010 => 0
[1,4,5,3,2] => 0011 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 2 = 1 + 1
[2,1] => 1 => 1 = 0 + 1
[1,2,3] => 00 => 3 = 2 + 1
[1,3,2] => 01 => 2 = 1 + 1
[2,1,3] => 10 => 1 = 0 + 1
[2,3,1] => 01 => 2 = 1 + 1
[3,1,2] => 10 => 1 = 0 + 1
[3,2,1] => 11 => 1 = 0 + 1
[1,2,3,4] => 000 => 4 = 3 + 1
[1,2,4,3] => 001 => 3 = 2 + 1
[1,3,2,4] => 010 => 2 = 1 + 1
[1,3,4,2] => 001 => 3 = 2 + 1
[1,4,2,3] => 010 => 2 = 1 + 1
[1,4,3,2] => 011 => 2 = 1 + 1
[2,1,3,4] => 100 => 1 = 0 + 1
[2,1,4,3] => 101 => 1 = 0 + 1
[2,3,1,4] => 010 => 2 = 1 + 1
[2,3,4,1] => 001 => 3 = 2 + 1
[2,4,1,3] => 010 => 2 = 1 + 1
[2,4,3,1] => 011 => 2 = 1 + 1
[3,1,2,4] => 100 => 1 = 0 + 1
[3,1,4,2] => 101 => 1 = 0 + 1
[3,2,1,4] => 110 => 1 = 0 + 1
[3,2,4,1] => 101 => 1 = 0 + 1
[3,4,1,2] => 010 => 2 = 1 + 1
[3,4,2,1] => 011 => 2 = 1 + 1
[4,1,2,3] => 100 => 1 = 0 + 1
[4,1,3,2] => 101 => 1 = 0 + 1
[4,2,1,3] => 110 => 1 = 0 + 1
[4,2,3,1] => 101 => 1 = 0 + 1
[4,3,1,2] => 110 => 1 = 0 + 1
[4,3,2,1] => 111 => 1 = 0 + 1
[1,2,3,4,5] => 0000 => 5 = 4 + 1
[1,2,3,5,4] => 0001 => 4 = 3 + 1
[1,2,4,3,5] => 0010 => 3 = 2 + 1
[1,2,4,5,3] => 0001 => 4 = 3 + 1
[1,2,5,3,4] => 0010 => 3 = 2 + 1
[1,2,5,4,3] => 0011 => 3 = 2 + 1
[1,3,2,4,5] => 0100 => 2 = 1 + 1
[1,3,2,5,4] => 0101 => 2 = 1 + 1
[1,3,4,2,5] => 0010 => 3 = 2 + 1
[1,3,4,5,2] => 0001 => 4 = 3 + 1
[1,3,5,2,4] => 0010 => 3 = 2 + 1
[1,3,5,4,2] => 0011 => 3 = 2 + 1
[1,4,2,3,5] => 0100 => 2 = 1 + 1
[1,4,2,5,3] => 0101 => 2 = 1 + 1
[1,4,3,2,5] => 0110 => 2 = 1 + 1
[1,4,3,5,2] => 0101 => 2 = 1 + 1
[1,4,5,2,3] => 0010 => 3 = 2 + 1
[1,4,5,3,2] => 0011 => 3 = 2 + 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: St000382
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => 2 = 1 + 1
[2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [3] => 3 = 2 + 1
[1,3,2] => [2,1] => 2 = 1 + 1
[2,1,3] => [1,2] => 1 = 0 + 1
[2,3,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [4] => 4 = 3 + 1
[1,2,4,3] => [3,1] => 3 = 2 + 1
[1,3,2,4] => [2,2] => 2 = 1 + 1
[1,3,4,2] => [3,1] => 3 = 2 + 1
[1,4,2,3] => [2,2] => 2 = 1 + 1
[1,4,3,2] => [2,1,1] => 2 = 1 + 1
[2,1,3,4] => [1,3] => 1 = 0 + 1
[2,1,4,3] => [1,2,1] => 1 = 0 + 1
[2,3,1,4] => [2,2] => 2 = 1 + 1
[2,3,4,1] => [3,1] => 3 = 2 + 1
[2,4,1,3] => [2,2] => 2 = 1 + 1
[2,4,3,1] => [2,1,1] => 2 = 1 + 1
[3,1,2,4] => [1,3] => 1 = 0 + 1
[3,1,4,2] => [1,2,1] => 1 = 0 + 1
[3,2,1,4] => [1,1,2] => 1 = 0 + 1
[3,2,4,1] => [1,2,1] => 1 = 0 + 1
[3,4,1,2] => [2,2] => 2 = 1 + 1
[3,4,2,1] => [2,1,1] => 2 = 1 + 1
[4,1,2,3] => [1,3] => 1 = 0 + 1
[4,1,3,2] => [1,2,1] => 1 = 0 + 1
[4,2,1,3] => [1,1,2] => 1 = 0 + 1
[4,2,3,1] => [1,2,1] => 1 = 0 + 1
[4,3,1,2] => [1,1,2] => 1 = 0 + 1
[4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [5] => 5 = 4 + 1
[1,2,3,5,4] => [4,1] => 4 = 3 + 1
[1,2,4,3,5] => [3,2] => 3 = 2 + 1
[1,2,4,5,3] => [4,1] => 4 = 3 + 1
[1,2,5,3,4] => [3,2] => 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => 3 = 2 + 1
[1,3,2,4,5] => [2,3] => 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => 2 = 1 + 1
[1,3,4,2,5] => [3,2] => 3 = 2 + 1
[1,3,4,5,2] => [4,1] => 4 = 3 + 1
[1,3,5,2,4] => [3,2] => 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => 3 = 2 + 1
[1,4,2,3,5] => [2,3] => 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [2,1,2] => 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [3,2] => 3 = 2 + 1
[1,4,5,3,2] => [3,1,1] => 3 = 2 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => 2 = 1 + 1
[2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [3] => 3 = 2 + 1
[1,3,2] => [2,1] => 1 = 0 + 1
[2,1,3] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => 1 = 0 + 1
[3,1,2] => [1,2] => 2 = 1 + 1
[3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [4] => 4 = 3 + 1
[1,2,4,3] => [3,1] => 1 = 0 + 1
[1,3,2,4] => [2,2] => 2 = 1 + 1
[1,3,4,2] => [3,1] => 1 = 0 + 1
[1,4,2,3] => [2,2] => 2 = 1 + 1
[1,4,3,2] => [2,1,1] => 1 = 0 + 1
[2,1,3,4] => [1,3] => 3 = 2 + 1
[2,1,4,3] => [1,2,1] => 1 = 0 + 1
[2,3,1,4] => [2,2] => 2 = 1 + 1
[2,3,4,1] => [3,1] => 1 = 0 + 1
[2,4,1,3] => [2,2] => 2 = 1 + 1
[2,4,3,1] => [2,1,1] => 1 = 0 + 1
[3,1,2,4] => [1,3] => 3 = 2 + 1
[3,1,4,2] => [1,2,1] => 1 = 0 + 1
[3,2,1,4] => [1,1,2] => 2 = 1 + 1
[3,2,4,1] => [1,2,1] => 1 = 0 + 1
[3,4,1,2] => [2,2] => 2 = 1 + 1
[3,4,2,1] => [2,1,1] => 1 = 0 + 1
[4,1,2,3] => [1,3] => 3 = 2 + 1
[4,1,3,2] => [1,2,1] => 1 = 0 + 1
[4,2,1,3] => [1,1,2] => 2 = 1 + 1
[4,2,3,1] => [1,2,1] => 1 = 0 + 1
[4,3,1,2] => [1,1,2] => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [5] => 5 = 4 + 1
[1,2,3,5,4] => [4,1] => 1 = 0 + 1
[1,2,4,3,5] => [3,2] => 2 = 1 + 1
[1,2,4,5,3] => [4,1] => 1 = 0 + 1
[1,2,5,3,4] => [3,2] => 2 = 1 + 1
[1,2,5,4,3] => [3,1,1] => 1 = 0 + 1
[1,3,2,4,5] => [2,3] => 3 = 2 + 1
[1,3,2,5,4] => [2,2,1] => 1 = 0 + 1
[1,3,4,2,5] => [3,2] => 2 = 1 + 1
[1,3,4,5,2] => [4,1] => 1 = 0 + 1
[1,3,5,2,4] => [3,2] => 2 = 1 + 1
[1,3,5,4,2] => [3,1,1] => 1 = 0 + 1
[1,4,2,3,5] => [2,3] => 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => 1 = 0 + 1
[1,4,3,2,5] => [2,1,2] => 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => 1 = 0 + 1
[1,4,5,2,3] => [3,2] => 2 = 1 + 1
[1,4,5,3,2] => [3,1,1] => 1 = 0 + 1
Description
The last part of an integer composition.
Matching statistic: St000678
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000745
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,2]]
=> 1 = 0 + 1
[2,1] => [[1],[2]]
=> 2 = 1 + 1
[1,2,3] => [[1,2,3]]
=> 1 = 0 + 1
[1,3,2] => [[1,2],[3]]
=> 1 = 0 + 1
[2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [[1,2],[3]]
=> 1 = 0 + 1
[3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,3,4] => [[1,2,3,4]]
=> 1 = 0 + 1
[1,2,4,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,3,2,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,3,4,2] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,4,2,3] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,4,3,2] => [[1,2],[3],[4]]
=> 1 = 0 + 1
[2,1,3,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[2,3,1,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[2,3,4,1] => [[1,2,3],[4]]
=> 1 = 0 + 1
[2,4,1,3] => [[1,2],[3,4]]
=> 1 = 0 + 1
[2,4,3,1] => [[1,2],[3],[4]]
=> 1 = 0 + 1
[3,1,2,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[3,2,1,4] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[3,4,2,1] => [[1,2],[3],[4]]
=> 1 = 0 + 1
[4,1,2,3] => [[1,3,4],[2]]
=> 2 = 1 + 1
[4,1,3,2] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1 = 0 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 1 = 0 + 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 0 + 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1 = 0 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 1 = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001050
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> 2 = 1 + 1
[2,1] => {{1,2}}
=> 1 = 0 + 1
[1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[1,3,2] => {{1},{2,3}}
=> 1 = 0 + 1
[2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[2,3,1] => {{1,2,3}}
=> 1 = 0 + 1
[3,1,2] => {{1,2,3}}
=> 1 = 0 + 1
[3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 0 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 2 = 1 + 1
[1,3,4,2] => {{1},{2,3,4}}
=> 1 = 0 + 1
[1,4,2,3] => {{1},{2,3,4}}
=> 1 = 0 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 1 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> 1 = 0 + 1
[2,3,1,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[2,3,4,1] => {{1,2,3,4}}
=> 1 = 0 + 1
[2,4,1,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[2,4,3,1] => {{1,2,4},{3}}
=> 2 = 1 + 1
[3,1,2,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[3,1,4,2] => {{1,2,3,4}}
=> 1 = 0 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[3,2,4,1] => {{1,3,4},{2}}
=> 1 = 0 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2 = 1 + 1
[3,4,2,1] => {{1,2,3,4}}
=> 1 = 0 + 1
[4,1,2,3] => {{1,2,3,4}}
=> 1 = 0 + 1
[4,1,3,2] => {{1,2,4},{3}}
=> 2 = 1 + 1
[4,2,1,3] => {{1,3,4},{2}}
=> 1 = 0 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[4,3,1,2] => {{1,2,3,4}}
=> 1 = 0 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 4 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1 = 0 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2 = 1 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1 = 0 + 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1 = 0 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2 = 1 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 3 = 2 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1 = 0 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2 = 1 + 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1 = 0 + 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 1 = 0 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2 = 1 + 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2 = 1 + 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 1 = 0 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3 = 2 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1 = 0 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2 = 1 + 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> 1 = 0 + 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
The following 240 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000759The smallest missing part in an integer partition. St000916The packing number of a graph. St000991The number of right-to-left minima of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. 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. St001829The common independence number of a graph. St000439The position of the first down step of a 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. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000090The variation of a composition. St000133The "bounce" of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000007The number of saliances of the permutation. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000335The difference of lower and upper interactions. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000717The number of ordinal summands of a poset. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000823The number of unsplittable factors of the set partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000917The open packing number of a graph. St000971The smallest closer of a set partition. St000993The multiplicity of the largest part of an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001481The minimal height of a peak of a Dyck path. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001828The Euler characteristic of a graph. St000258The burning number of a graph. St000738The first entry in the last row of a standard tableau. St001340The cardinality of a minimal non-edge isolating set of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000377The dinv defect of an integer partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001484The number of singletons of an integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000181The number of connected components of the Hasse diagram for the poset. St000940The number of characters of the symmetric group whose value on the partition is zero. St000658The number of rises of length 2 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000938The number of zeros of the symmetric group character corresponding to the partition. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000137The Grundy value of an integer partition. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000260The radius of a connected graph. St001651The Frankl number of a lattice. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001877Number of indecomposable injective modules with projective dimension 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000454The largest eigenvalue of a graph if it is integral. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000936The number of even values of the symmetric group character corresponding to the partition. St000699The toughness times the least common multiple of 1,. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000455The second largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000338The number of pixed points of a permutation. St000732The number of double deficiencies of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001561The value of the elementary symmetric function evaluated at 1. St001961The sum of the greatest common divisors of all pairs of parts. St001862The number of crossings of a signed permutation. St001720The minimal length of a chain of small intervals in a lattice. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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