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Statistic identifier: St000618
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Collection: Integer partitions
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Description: The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
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References: [1] Stembridge, J. R. Canonical bases and self-evacuating tableaux [[MathSciNet:1387685]]
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Code:
def statistic_alt(la):
chi = SymmetricGroupRepresentation(la)
r = abs(ZZ(chi.to_character()(Permutation(range(la.size(),0,-1)))))
assert r==statistic(la)
return r
def statistic(la):
n = la.size()
la = la + [0]*(n-len(la))
E_la = sorted([la[j] + n-j-1 for j in range(n) if is_even(la[j] + n-j-1)], reverse=True)
rE = len(E_la)
la_e = [ZZ(E_la[i]/2-rE+i+1) for i in range(rE)]
O_la = sorted([la[j] + n-j-2 for j in range(n) if is_even(la[j] +n-j-2)], reverse=True)
rO = len(O_la)
la_o = [ZZ(O_la[i]/2-rO+i+1) for i in range(rO)]
if abs(rE - rO)<=1:
return (binomial(n//2, sum(la_e))*
StandardTableaux(la_e).cardinality()*
StandardTableaux(la_o).cardinality())
else:
return 0
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 0
[1,1,1] => 1
[4] => 1
[3,1] => 1
[2,2] => 2
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 1
[4,1] => 0
[3,2] => 1
[3,1,1] => 2
[2,2,1] => 1
[2,1,1,1] => 0
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 1
[4,2] => 3
[4,1,1] => 2
[3,3] => 3
[3,2,1] => 0
[3,1,1,1] => 2
[2,2,2] => 3
[2,2,1,1] => 3
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 0
[5,2] => 2
[5,1,1] => 3
[4,3] => 0
[4,2,1] => 1
[4,1,1,1] => 0
[3,3,1] => 3
[3,2,2] => 3
[3,2,1,1] => 1
[3,1,1,1,1] => 3
[2,2,2,1] => 0
[2,2,1,1,1] => 2
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 1
[6,2] => 4
[6,1,1] => 3
[5,3] => 4
[5,2,1] => 0
[5,1,1,1] => 3
[4,4] => 6
[4,3,1] => 2
[4,2,2] => 8
[4,2,1,1] => 6
[4,1,1,1,1] => 3
[3,3,2] => 6
[3,3,1,1] => 8
[3,2,2,1] => 2
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 3
[2,2,2,2] => 6
[2,2,2,1,1] => 4
[2,2,1,1,1,1] => 4
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 0
[7,2] => 3
[7,1,1] => 4
[6,3] => 0
[6,2,1] => 1
[6,1,1,1] => 0
[5,4] => 2
[5,3,1] => 6
[5,2,2] => 8
[5,2,1,1] => 3
[5,1,1,1,1] => 6
[4,4,1] => 4
[4,3,2] => 0
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 3
[4,1,1,1,1,1] => 0
[3,3,3] => 6
[3,3,2,1] => 0
[3,3,1,1,1] => 8
[3,2,2,2] => 4
[3,2,2,1,1] => 6
[3,2,1,1,1,1] => 1
[3,1,1,1,1,1,1] => 4
[2,2,2,2,1] => 2
[2,2,2,1,1,1] => 0
[2,2,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 1
[8,2] => 5
[8,1,1] => 4
[7,3] => 5
[7,2,1] => 0
[7,1,1,1] => 4
[6,4] => 10
[6,3,1] => 5
[6,2,2] => 15
[6,2,1,1] => 10
[6,1,1,1,1] => 6
[5,5] => 10
[5,4,1] => 0
[5,3,2] => 10
[5,3,1,1] => 15
[5,2,2,1] => 5
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 6
[4,4,2] => 20
[4,4,1,1] => 20
[4,3,3] => 10
[4,3,2,1] => 0
[4,3,1,1,1] => 5
[4,2,2,2] => 20
[4,2,2,1,1] => 15
[4,2,1,1,1,1] => 10
[4,1,1,1,1,1,1] => 4
[3,3,3,1] => 10
[3,3,2,2] => 20
[3,3,2,1,1] => 10
[3,3,1,1,1,1] => 15
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 5
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 4
[2,2,2,2,2] => 10
[2,2,2,2,1,1] => 10
[2,2,2,1,1,1,1] => 5
[2,2,1,1,1,1,1,1] => 5
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 1
[10,1] => 0
[9,2] => 4
[9,1,1] => 5
[8,3] => 0
[8,2,1] => 1
[8,1,1,1] => 0
[7,4] => 5
[7,3,1] => 10
[7,2,2] => 15
[7,2,1,1] => 6
[7,1,1,1,1] => 10
[6,5] => 0
[6,4,1] => 5
[6,3,2] => 0
[6,3,1,1] => 0
[6,2,2,1] => 0
[6,2,1,1,1] => 4
[6,1,1,1,1,1] => 0
[5,5,1] => 10
[5,4,2] => 10
[5,4,1,1] => 5
[5,3,3] => 20
[5,3,2,1] => 0
[5,3,1,1,1] => 20
[5,2,2,2] => 15
[5,2,2,1,1] => 20
[5,2,1,1,1,1] => 4
[5,1,1,1,1,1,1] => 10
[4,4,3] => 10
[4,4,2,1] => 0
[4,4,1,1,1] => 15
[4,3,3,1] => 0
[4,3,2,2] => 0
[4,3,2,1,1] => 0
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 5
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 10
[3,3,3,1,1] => 20
[3,3,2,2,1] => 10
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => 15
[3,2,2,2,2] => 10
[3,2,2,2,1,1] => 5
[3,2,2,1,1,1,1] => 10
[3,2,1,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1,1] => 5
[2,2,2,2,2,1] => 0
[2,2,2,2,1,1,1] => 5
[2,2,2,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 1
[11,1] => 1
[10,2] => 6
[10,1,1] => 5
[9,3] => 6
[9,2,1] => 0
[9,1,1,1] => 5
[8,4] => 15
[8,3,1] => 9
[8,2,2] => 24
[8,2,1,1] => 15
[8,1,1,1,1] => 10
[7,5] => 15
[7,4,1] => 0
[7,3,2] => 15
[7,3,1,1] => 24
[7,2,2,1] => 9
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 10
[6,6] => 20
[6,5,1] => 5
[6,4,2] => 45
[6,4,1,1] => 40
[6,3,3] => 30
[6,3,2,1] => 0
[6,3,1,1,1] => 16
[6,2,2,2] => 45
[6,2,2,1,1] => 36
[6,2,1,1,1,1] => 20
[6,1,1,1,1,1,1] => 10
[5,5,2] => 40
[5,5,1,1] => 45
[5,4,3] => 0
[5,4,2,1] => 5
[5,4,1,1,1] => 0
[5,3,3,1] => 30
[5,3,2,2] => 45
[5,3,2,1,1] => 20
[5,3,1,1,1,1] => 36
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 16
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 10
[4,4,4] => 30
[4,4,3,1] => 30
[4,4,2,2] => 80
[4,4,2,1,1] => 45
[4,4,1,1,1,1] => 45
[4,3,3,2] => 30
[4,3,3,1,1] => 30
[4,3,2,2,1] => 5
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 9
[4,2,2,2,2] => 45
[4,2,2,2,1,1] => 40
[4,2,2,1,1,1,1] => 24
[4,2,1,1,1,1,1,1] => 15
[4,1,1,1,1,1,1,1,1] => 5
[3,3,3,3] => 30
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 30
[3,3,2,2,2] => 40
[3,3,2,2,1,1] => 45
[3,3,2,1,1,1,1] => 15
[3,3,1,1,1,1,1,1] => 24
[3,2,2,2,2,1] => 5
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 9
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 5
[2,2,2,2,2,2] => 20
[2,2,2,2,2,1,1] => 15
[2,2,2,2,1,1,1,1] => 15
[2,2,2,1,1,1,1,1,1] => 6
[2,2,1,1,1,1,1,1,1,1] => 6
[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
-----------------------------------------------------------------------------
Created: Sep 26, 2016 at 23:00 by Martin Rubey
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Last Updated: Sep 26, 2016 at 23:00 by Martin Rubey