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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000878
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St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => -1
1 => 1
00 => -2
01 => 0
10 => 0
11 => 2
000 => -3
001 => -1
010 => -1
011 => 1
100 => -1
101 => 1
110 => 1
111 => 3
0000 => -4
0001 => -2
0010 => -2
0011 => 0
0100 => -2
0101 => 0
0110 => 0
0111 => 2
1000 => -2
1001 => 0
1010 => 0
1011 => 2
1100 => 0
1101 => 2
1110 => 2
1111 => 4
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000145
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> [2]
=> 1
1 => [1,1] => [[1,1],[]]
=> [1,1]
=> -1
00 => [3] => [[3],[]]
=> [3]
=> 2
01 => [2,1] => [[2,2],[1]]
=> [2,2]
=> 0
10 => [1,2] => [[2,1],[]]
=> [2,1]
=> 0
11 => [1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> -2
000 => [4] => [[4],[]]
=> [4]
=> 3
001 => [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
010 => [2,2] => [[3,2],[1]]
=> [3,2]
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> -1
100 => [1,3] => [[3,1],[]]
=> [3,1]
=> 1
101 => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> -1
110 => [1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> -1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> -3
0000 => [5] => [[5],[]]
=> [5]
=> 4
0001 => [4,1] => [[4,4],[3]]
=> [4,4]
=> 2
0010 => [3,2] => [[4,3],[2]]
=> [4,3]
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 0
0100 => [2,3] => [[4,2],[1]]
=> [4,2]
=> 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 0
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> -2
1000 => [1,4] => [[4,1],[]]
=> [4,1]
=> 2
1001 => [1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> 0
1010 => [1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> -2
1100 => [1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> -2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> -2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> -4
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000512
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000514
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000514: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000514: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
Description
The number of invariant simple graphs when acting with a permutation of given cycle type.
Matching statistic: St000515
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,0,0,0,0,4}
Description
The number of invariant set partitions when acting with a permutation of given cycle type.
Matching statistic: St000936
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
Description
The number of even values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
Matching statistic: St000938
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000940
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St000941
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 13%●distinct values known / distinct values provided: 11%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,-2,-2,0,0,2,2,2,2,4}
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St000997
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000997: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 22%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000997: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 22%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {-1,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {-2,0,0,2}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {-2,0,0,2}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {-3,-1,-1,-1,1,1,1,3}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> -2
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> -2
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {-4,-2,-2,0,0,0,0,0,0,2,2,4}
Description
The even-odd crank of an integer partition.
This is the largest even part minus the number of odd parts.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice.
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