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Identifier

St000202: Cores ⟶ ℤ

Values

([2],3) => 3

([1,1],3) => 3

([2],4) => 3

([1,1],4) => 3

([2],5) => 3

([1,1],5) => 3

([2],6) => 3

([1,1],6) => 3

([3,1],3) => 5

([2,1,1],3) => 5

([3],4) => 4

([2,1],4) => 5

([1,1,1],4) => 4

([3],5) => 4

([2,1],5) => 5

([1,1,1],5) => 4

([3],6) => 4

([2,1],6) => 5

([1,1,1],6) => 4

([4,2],3) => 6

([3,1,1],3) => 7

([2,2,1,1],3) => 6

([4,1],4) => 7

([2,2],4) => 6

([3,1,1],4) => 8

([2,1,1,1],4) => 7

([4],5) => 5

([3,1],5) => 7

([2,2],5) => 6

([2,1,1],5) => 7

([1,1,1,1],5) => 5

([4],6) => 5

([3,1],6) => 7

([2,2],6) => 6

([2,1,1],6) => 7

([1,1,1,1],6) => 5

([5,3,1],3) => 9

([4,2,1,1],3) => 10

([3,2,2,1,1],3) => 9

([5,2],4) => 9

([4,1,1],4) => 10

([3,2,1],4) => 10

([3,1,1,1],4) => 10

([2,2,1,1,1],4) => 9

([5,1],5) => 9

([3,2],5) => 9

([4,1,1],5) => 11

([2,2,1],5) => 9

([3,1,1,1],5) => 11

([2,1,1,1,1],5) => 9

([5],6) => 6

([4,1],6) => 9

([3,2],6) => 9

([3,1,1],6) => 10

([2,2,1],6) => 9

([2,1,1,1],6) => 9

([1,1,1,1,1],6) => 6

([5,3,1,1],3) => 12

([4,2,2,1,1],3) => 12

([6,3],4) => 10

([5,2,1],4) => 14

([4,1,1,1],4) => 13

([4,2,2],4) => 13

([3,3,1,1],4) => 13

([3,2,1,1,1],4) => 14

([2,2,2,1,1,1],4) => 10

([6,2],5) => 12

([5,1,1],5) => 13

([3,3],5) => 10

([4,2,1],5) => 15

([4,1,1,1],5) => 14

([2,2,2],5) => 10

([3,2,1,1],5) => 15

([3,1,1,1,1],5) => 13

([2,2,1,1,1,1],5) => 12

([6,1],6) => 11

([4,2],6) => 12

([5,1,1],6) => 14

([3,3],6) => 10

([3,2,1],6) => 14

([4,1,1,1],6) => 15

([2,2,2],6) => 10

([2,2,1,1],6) => 12

([3,1,1,1,1],6) => 14

([2,1,1,1,1,1],6) => 11

([7,2],6) => 15

([6,1,1],6) => 16

([4,3],6) => 14

([5,2,1],6) => 20

([5,1,1,1],6) => 18

([3,3,1],6) => 16

([3,2,2],6) => 16

([4,2,1,1],6) => 22

([4,1,1,1,1],6) => 18

([2,2,2,1],6) => 14

([3,2,1,1,1],6) => 20

([3,1,1,1,1,1],6) => 16

([2,2,1,1,1,1,1],6) => 15

([1],2) => 2

([2,1],2) => 3

([3,2,1],2) => 4

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Description

The number of k-cores contained in the k-core.
Let $\lambda$ and $\mu$ be $k$-cores. Then $\lambda$ contains $\mu$ if and only if the Ferrers diagram of $\lambda$ contains the diagram of $\mu$. Each nonempty core trivially contains two other cores, the empty core and itself. The poset corresponding to containment is Young's lattice restricted to cores [1].

References

[1] Lam, T., Lapointe, L., Morse, J., Schilling, A., Shimozono, M., Zabrocki, M. $k$-Schur functions and affine Schubert calculus arXiv:1301.3569

Code

def statistic(c):
    a = 0
    for m in range(c.size()+1):
        for d in Cores(c.k(), size=m):
            if c.contains(d):
                a += 1
    return a

Created

May 16, 2014 at 14:46 by Patrick Bjerke

Updated

Nov 16, 2017 at 11:39 by Martin Rubey