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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000651
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Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ 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]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St000956
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ 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] => 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St001090
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001090: Permutations ⟶ 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] => 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St001207
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ 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] => 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001418
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001418: 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]
=> [1,0,1,0]
=> 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000567
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2]
=> [2]
=> []
=> ?
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 1
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2]
=> [2]
=> []
=> ?
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 1
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[6,2]
=> [2]
=> []
=> ?
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 1
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,2]
=> [2]
=> []
=> ?
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[3,3,3]
=> [3,3]
=> [3]
=> []
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,2]
=> [2]
=> []
=> ?
=> ? = 1
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[9,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[7,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[10,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[8,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[11,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[9,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[12,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[10,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[13,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[11,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000620
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2]
=> [2]
=> []
=> ?
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 1
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2]
=> [2]
=> []
=> ?
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 1
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[6,2]
=> [2]
=> []
=> ?
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 1
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,2]
=> [2]
=> []
=> ?
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[3,3,3]
=> [3,3]
=> [3]
=> []
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,2]
=> [2]
=> []
=> ?
=> ? = 1
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[9,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[7,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[10,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[8,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[11,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[9,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[12,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[10,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[13,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[11,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is [[St000621]].
Matching statistic: St000929
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2]
=> [2]
=> []
=> ?
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 1
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2]
=> [2]
=> []
=> ?
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 1
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[6,2]
=> [2]
=> []
=> ?
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 1
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,2]
=> [2]
=> []
=> ?
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[3,3,3]
=> [3,3]
=> [3]
=> []
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,2]
=> [2]
=> []
=> ?
=> ? = 1
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[9,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[7,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[10,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[8,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[11,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[9,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[12,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[10,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[13,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[11,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001099
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2]
=> [2]
=> []
=> ?
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 1
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2]
=> [2]
=> []
=> ?
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 1
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[6,2]
=> [2]
=> []
=> ?
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 1
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,2]
=> [2]
=> []
=> ?
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[3,3,3]
=> [3,3]
=> [3]
=> []
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,2]
=> [2]
=> []
=> ?
=> ? = 1
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[9,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[7,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[10,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[8,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[11,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[9,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[12,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[10,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[13,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[11,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
Description
The 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.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
Matching statistic: St001100
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001100: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001100: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2]
=> [2]
=> []
=> ?
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2]
=> [2]
=> []
=> ?
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[3,3]
=> [3]
=> []
=> ?
=> ? = 1
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2]
=> [2]
=> []
=> ?
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[4,3]
=> [3]
=> []
=> ?
=> ? = 1
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[6,2]
=> [2]
=> []
=> ?
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[5,3]
=> [3]
=> []
=> ?
=> ? = 1
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,2]
=> [2]
=> []
=> ?
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[6,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? = 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? = 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? = 2
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? = 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> []
=> ? = 3
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? = 2
[3,3,3]
=> [3,3]
=> [3]
=> []
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,2]
=> [2]
=> []
=> ?
=> ? = 1
[8,1,1]
=> [1,1]
=> [1]
=> []
=> ? = 0
[7,3]
=> [3]
=> []
=> ?
=> ? = 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[7,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[8,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[9,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[7,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[10,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[8,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[11,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[9,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[12,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[10,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[13,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[11,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
Description
The 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.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, whose generating function is the reversal of $f^{(-1)}(x) = 1+2x-\exp(x)$, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
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