Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000254
(load all 53 compositions to match this statistic)
Mapping
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => {{1},{2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1,2}}
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => {{1},{2},{3}}
 => 0
{{1,2},{3}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1,2,3}}
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3},{4}}
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 1
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 1
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 1
{{1,2},{3},{4}}
 => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 1
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => 2
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => 1
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => 1
{{1},{2,3},{4}}
 => {{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => 1
{{1},{2},{3,4}}
 => {{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 1
Description
The nesting number of a set partition. This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St001859
(load all 38 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1},{2}}
 => [1,2] => 0
{{1},{2}}
 => {{1,2}}
 => [2,1] => 1
{{1,2,3}}
 => {{1},{2},{3}}
 => [1,2,3] => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => [2,3,1] => 1
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 1
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 1
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 1
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 1
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
{{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 1
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 1
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 1
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 1
Description
The number of factors of the Stanley symmetric function associated with a permutation. For example, the Stanley symmetric function of $\pi=321645$ equals $20 m_{1,1,1,1,1} + 11 m_{2,1,1,1} + 6 m_{2,2,1} + 4 m_{3,1,1} + 2 m_{3,2} + m_{4,1} = (m_{1,1} + m_{2})(2 m_{1,1,1} + m_{2,1}).$
Matching statistic: St000162
(load all 4 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => [3,1,2] => 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => [2,1,3] => 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => [2,3,1] => 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [3,2,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => [3,1,2,4] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => [4,1,3,2] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => [3,1,4,2] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [1,4,2,3] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => [2,4,3,1] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => [2,3,4,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => [3,2,1,4] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => [3,4,2,1] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => [4,2,3,1] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [3,2,4,1] => 1
Description
The number of nontrivial cycles in the cycle decomposition of a permutation. This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St000245
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,3,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [2,4,1,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,4,3,2] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [2,4,3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,4,3,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000253
(load all 21 compositions to match this statistic)
Mapping
Mp00219: Set partitions —inverse Yip⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1,2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1,2}}
 => {{1,2}}
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => {{1,2},{3}}
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 1
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => {{1,3},{2,4}}
 => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1,3,4},{2}}
 => 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => 1
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 1
Description
The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000374
(load all 4 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1},{2}}
 => [1,2] => [1,2] => 0
{{1},{2}}
 => {{1,2}}
 => [2,1] => [2,1] => 1
{{1,2,3}}
 => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => [2,3,1] => [2,3,1] => 1
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 1
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 1
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => 1
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => 1
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => [3,4,2,1] => 1
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => 1
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 1
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => 1
Description
The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
Matching statistic: St000390
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [1,2] => 0 => 0
{{1},{2}}
 => [1,2] => [2,1] => 1 => 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 00 => 0
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 01 => 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 10 => 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 11 => 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 01 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 000 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 001 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 010 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 011 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 001 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 100 => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 001 => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 101 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 110 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 011 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 011 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 010 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 111 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 011 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 001 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000672
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,3,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [2,4,1,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,4,3,2] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [2,4,3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,4,3,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1
Description
The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is $$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$ for some $(r,a,b)$. This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000834
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,3,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [2,4,1,3] => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,4,3,2] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [2,4,3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,4,3,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1
Description
The number of right outer peaks of a permutation. A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000996
(load all 2 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => {{1},{2}}
 => [1,2] => [1,2] => 0
{{1},{2}}
 => {{1,2}}
 => [2,1] => [2,1] => 1
{{1,2,3}}
 => {{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => [2,3,1] => [3,2,1] => 1
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 1
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 1
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => 1
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => 1
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 1
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => [4,3,2,1] => 1
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => [4,3,2,1] => 1
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => [4,3,2,1] => 1
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 1
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => [4,2,3,1] => 1
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => [4,2,3,1] => 1
Description
The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001261The Castelnuovo-Mumford regularity of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. St001651The Frankl number of a lattice. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001060The distinguishing index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.