Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000506
Mapping
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 1
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 2
[3,1,1]
 => 2
[2,2,1]
 => 2
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 1
[4,2]
 => 3
[4,1,1]
 => 3
[3,3]
 => 2
[3,2,1]
 => 6
[3,1,1,1]
 => 4
[2,2,2]
 => 2
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 0
[6,1]
 => 1
[5,2]
 => 4
[5,1,1]
 => 4
[4,3]
 => 5
[4,2,1]
 => 12
[4,1,1,1]
 => 7
[3,3,1]
 => 8
[3,2,2]
 => 8
[3,2,1,1]
 => 14
[3,1,1,1,1]
 => 6
[2,2,2,1]
 => 6
[2,2,1,1,1]
 => 6
[2,1,1,1,1,1]
 => 3
[1,1,1,1,1,1,1]
 => 0
[8]
 => 0
[7,1]
 => 1
[6,2]
 => 5
[6,1,1]
 => 5
[5,3]
 => 9
[5,2,1]
 => 20
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St000621
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => ? = 0
[2]
 => [1,1]
 => 0
[1,1]
 => [2]
 => 1
[3]
 => [1,1,1]
 => 0
[2,1]
 => [2,1]
 => 1
[1,1,1]
 => [3]
 => 0
[4]
 => [1,1,1,1]
 => 0
[3,1]
 => [2,1,1]
 => 1
[2,2]
 => [2,2]
 => 1
[2,1,1]
 => [3,1]
 => 1
[1,1,1,1]
 => [4]
 => 1
[5]
 => [1,1,1,1,1]
 => 0
[4,1]
 => [2,1,1,1]
 => 1
[3,2]
 => [2,2,1]
 => 2
[3,1,1]
 => [3,1,1]
 => 2
[2,2,1]
 => [3,2]
 => 2
[2,1,1,1]
 => [4,1]
 => 2
[1,1,1,1,1]
 => [5]
 => 0
[6]
 => [1,1,1,1,1,1]
 => 0
[5,1]
 => [2,1,1,1,1]
 => 1
[4,2]
 => [2,2,1,1]
 => 3
[4,1,1]
 => [3,1,1,1]
 => 3
[3,3]
 => [2,2,2]
 => 2
[3,2,1]
 => [3,2,1]
 => 6
[3,1,1,1]
 => [4,1,1]
 => 4
[2,2,2]
 => [3,3]
 => 2
[2,2,1,1]
 => [4,2]
 => 4
[2,1,1,1,1]
 => [5,1]
 => 2
[1,1,1,1,1,1]
 => [6]
 => 1
[7]
 => [1,1,1,1,1,1,1]
 => 0
[6,1]
 => [2,1,1,1,1,1]
 => 1
[5,2]
 => [2,2,1,1,1]
 => 4
[5,1,1]
 => [3,1,1,1,1]
 => 4
[4,3]
 => [2,2,2,1]
 => 5
[4,2,1]
 => [3,2,1,1]
 => 12
[4,1,1,1]
 => [4,1,1,1]
 => 7
[3,3,1]
 => [3,2,2]
 => 8
[3,2,2]
 => [3,3,1]
 => 8
[3,2,1,1]
 => [4,2,1]
 => 14
[3,1,1,1,1]
 => [5,1,1]
 => 6
[2,2,2,1]
 => [4,3]
 => 6
[2,2,1,1,1]
 => [5,2]
 => 6
[2,1,1,1,1,1]
 => [6,1]
 => 3
[1,1,1,1,1,1,1]
 => [7]
 => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => 0
[7,1]
 => [2,1,1,1,1,1,1]
 => 1
[6,2]
 => [2,2,1,1,1,1]
 => 5
[6,1,1]
 => [3,1,1,1,1,1]
 => 5
[5,3]
 => [2,2,2,1,1]
 => 9
[5,2,1]
 => [3,2,1,1,1]
 => 20
[5,1,1,1]
 => [4,1,1,1,1]
 => 11
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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 even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].