Your data matches 82 different statistics following compositions of up to 3 maps.
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Mp00170: Permutations to signed permutationSigned permutations
Mp00244: Signed permutations barSigned permutations
Mp00166: Signed permutations even cycle typeInteger partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 3
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 3
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 3
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 3
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 3
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 3
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 3
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The number of characters of the symmetric group whose value on the partition is positive.
Mp00108: Permutations cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000477: Integer partitions ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? = 1 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? = 1 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? = 1 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[1,5,4,2,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[1,5,4,3,2] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[2,1,3,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,1,4,3,5] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,1,4,5,3] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[2,1,5,3,4] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[2,1,5,4,3] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,3,1,5,4] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,3,5,4,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,4,1,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,4,3,5,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,4,5,1,3] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[2,5,1,4,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,5,3,1,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[2,5,4,3,1] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,1,2,5,4] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,1,4,2,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,1,5,4,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,2,1,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[3,2,1,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[3,2,4,5,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,2,5,1,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,4,1,2,5] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[3,4,1,5,2] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,4,2,1,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,4,5,2,1] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,5,1,2,4] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[3,5,1,4,2] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[3,5,2,4,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,5,4,1,2] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[4,1,2,3,5] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[4,1,3,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[4,1,5,2,3] => [3,2]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[4,2,1,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[4,2,3,1,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
[5,2,3,4,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 1 - 1
Description
The weight of a partition according to Alladi.
Matching statistic: St000207
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000207: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000208: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000667
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000667: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000755
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000755: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St001389
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001389: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001527
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001527: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.
Matching statistic: St001571
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001571: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 2
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 2
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 2
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 2
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 2
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St000319
Mp00108: Permutations cycle typeInteger partitions
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 77%distinct values known / distinct values provided: 67%
Values
[2,1] => [2]
=> [2]
=> []
=> ? = 1 - 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? = 1 - 1
[2,1,3] => [2,1]
=> [3]
=> []
=> ? = 1 - 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? = 1 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? = 3 - 1
[2,3,4,1] => [4]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[2,4,1,3] => [4]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[3,1,4,2] => [4]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? = 3 - 1
[3,4,2,1] => [4]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[4,1,2,3] => [4]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
[4,3,1,2] => [4]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? = 3 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[1,3,4,5,2] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,3,5,2,4] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,4,2,5,3] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[1,4,5,3,2] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,5,2,3,4] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,4,2,3] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[1,5,4,3,2] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[2,1,4,5,3] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[2,1,5,3,4] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[2,1,5,4,3] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[2,3,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[2,3,4,1,5] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[2,3,5,4,1] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[2,4,1,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[2,4,3,5,1] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[2,4,5,1,3] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[2,5,1,4,3] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[2,5,3,1,4] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[2,5,4,3,1] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[3,1,2,5,4] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[3,1,4,2,5] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[3,1,5,4,2] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[3,2,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[3,2,4,5,1] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[3,2,5,1,4] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[3,4,1,2,5] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[3,4,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[3,4,2,1,5] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[3,4,5,2,1] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[3,5,1,2,4] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[3,5,1,4,2] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[3,5,2,4,1] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[3,5,4,1,2] => [3,2]
=> [4,1]
=> [1]
=> 0 = 1 - 1
[4,1,2,3,5] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[4,1,3,5,2] => [4,1]
=> [3,2]
=> [2]
=> 1 = 2 - 1
[4,2,5,1,3] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[4,3,2,1,5] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[4,5,3,1,2] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[5,2,4,3,1] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[5,3,2,4,1] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
[5,4,3,2,1] => [2,2,1]
=> [5]
=> []
=> ? = 3 - 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. 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. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001754The number of tolerances of a finite lattice. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St000422The energy of a graph, if it is integral. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000264The girth of a graph, which is not a tree. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial 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. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. 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. St001568The smallest positive integer that does not appear twice in the partition. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.