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Schur Q-polynomials
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Polynomials
$n$
$\lambda$ 
$Q_\lambda(x_1,\ldots,x_n)$
2
2:
2*x1^2 + 4*x1*x2 + 2*x2^2
2
3:
2*x1^3 + 4*x1^2*x2 + 4*x1*x2^2 + 2*x2^3
2
2, 1:
4*x1^2*x2 + 4*x1*x2^2
2
4:
2*x1^4 + 4*x1^3*x2 + 4*x1^2*x2^2 + 4*x1*x2^3 + 2*x2^4
2
3, 1:
4*x1^3*x2 + 8*x1^2*x2^2 + 4*x1*x2^3
2
5:
2*x1^5 + 4*x1^4*x2 + 4*x1^3*x2^2 + 4*x1^2*x2^3 + 4*x1*x2^4 + 2*x2^5
2
4, 1:
4*x1^4*x2 + 8*x1^3*x2^2 + 8*x1^2*x2^3 + 4*x1*x2^4
2
3, 2:
4*x1^3*x2^2 + 4*x1^2*x2^3
2
6:
2*x1^6 + 4*x1^5*x2 + 4*x1^4*x2^2 + 4*x1^3*x2^3 + 4*x1^2*x2^4 + 4*x1*x2^5 + 2*x2^6
2
5, 1:
4*x1^5*x2 + 8*x1^4*x2^2 + 8*x1^3*x2^3 + 8*x1^2*x2^4 + 4*x1*x2^5
2
4, 2:
4*x1^4*x2^2 + 8*x1^3*x2^3 + 4*x1^2*x2^4
3
2:
2*x1^2 + 4*x1*x2 + 2*x2^2 + 4*x1*x3 + 4*x2*x3 + 2*x3^2
3
3:
2*x1^3 + 4*x1^2*x2 + 4*x1*x2^2 + 2*x2^3 + 4*x1^2*x3 + 8*x1*x2*x3 + 4*x2^2*x3 + 4*x1*x3^2 + 4*x2*x3^2 + 2*x3^3
3
2, 1:
4*x1^2*x2 + 4*x1*x2^2 + 4*x1^2*x3 + 8*x1*x2*x3 + 4*x2^2*x3 + 4*x1*x3^2 + 4*x2*x3^2
3
4:
2*x1^4 + 4*x1^3*x2 + 4*x1^2*x2^2 + 4*x1*x2^3 + 2*x2^4 + 4*x1^3*x3 + 8*x1^2*x2*x3 + 8*x1*x2^2*x3 + 4*x2^3*x3 + 4*x1^2*x3^2 + 8*x1*x2*x3^2 + 4*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + 2*x3^4
3
3, 1:
4*x1^3*x2 + 8*x1^2*x2^2 + 4*x1*x2^3 + 4*x1^3*x3 + 16*x1^2*x2*x3 + 16*x1*x2^2*x3 + 4*x2^3*x3 + 8*x1^2*x3^2 + 16*x1*x2*x3^2 + 8*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3
3
5:
2*x1^5 + 4*x1^4*x2 + 4*x1^3*x2^2 + 4*x1^2*x2^3 + 4*x1*x2^4 + 2*x2^5 + 4*x1^4*x3 + 8*x1^3*x2*x3 + 8*x1^2*x2^2*x3 + 8*x1*x2^3*x3 + 4*x2^4*x3 + 4*x1^3*x3^2 + 8*x1^2*x2*x3^2 + 8*x1*x2^2*x3^2 + 4*x2^3*x3^2 + 4*x1^2*x3^3 + 8*x1*x2*x3^3 + 4*x2^2*x3^3 + 4*x1*x3^4 + 4*x2*x3^4 + 2*x3^5
3
4, 1:
4*x1^4*x2 + 8*x1^3*x2^2 + 8*x1^2*x2^3 + 4*x1*x2^4 + 4*x1^4*x3 + 16*x1^3*x2*x3 + 24*x1^2*x2^2*x3 + 16*x1*x2^3*x3 + 4*x2^4*x3 + 8*x1^3*x3^2 + 24*x1^2*x2*x3^2 + 24*x1*x2^2*x3^2 + 8*x2^3*x3^2 + 8*x1^2*x3^3 + 16*x1*x2*x3^3 + 8*x2^2*x3^3 + 4*x1*x3^4 + 4*x2*x3^4
3
3, 2:
4*x1^3*x2^2 + 4*x1^2*x2^3 + 8*x1^3*x2*x3 + 16*x1^2*x2^2*x3 + 8*x1*x2^3*x3 + 4*x1^3*x3^2 + 16*x1^2*x2*x3^2 + 16*x1*x2^2*x3^2 + 4*x2^3*x3^2 + 4*x1^2*x3^3 + 8*x1*x2*x3^3 + 4*x2^2*x3^3
3
6:
2*x1^6 + 4*x1^5*x2 + 4*x1^4*x2^2 + 4*x1^3*x2^3 + 4*x1^2*x2^4 + 4*x1*x2^5 + 2*x2^6 + 4*x1^5*x3 + 8*x1^4*x2*x3 + 8*x1^3*x2^2*x3 + 8*x1^2*x2^3*x3 + 8*x1*x2^4*x3 + 4*x2^5*x3 + 4*x1^4*x3^2 + 8*x1^3*x2*x3^2 + 8*x1^2*x2^2*x3^2 + 8*x1*x2^3*x3^2 + 4*x2^4*x3^2 + 4*x1^3*x3^3 + 8*x1^2*x2*x3^3 + 8*x1*x2^2*x3^3 + 4*x2^3*x3^3 + 4*x1^2*x3^4 + 8*x1*x2*x3^4 + 4*x2^2*x3^4 + 4*x1*x3^5 + 4*x2*x3^5 + 2*x3^6
3
5, 1:
4*x1^5*x2 + 8*x1^4*x2^2 + 8*x1^3*x2^3 + 8*x1^2*x2^4 + 4*x1*x2^5 + 4*x1^5*x3 + 16*x1^4*x2*x3 + 24*x1^3*x2^2*x3 + 24*x1^2*x2^3*x3 + 16*x1*x2^4*x3 + 4*x2^5*x3 + 8*x1^4*x3^2 + 24*x1^3*x2*x3^2 + 32*x1^2*x2^2*x3^2 + 24*x1*x2^3*x3^2 + 8*x2^4*x3^2 + 8*x1^3*x3^3 + 24*x1^2*x2*x3^3 + 24*x1*x2^2*x3^3 + 8*x2^3*x3^3 + 8*x1^2*x3^4 + 16*x1*x2*x3^4 + 8*x2^2*x3^4 + 4*x1*x3^5 + 4*x2*x3^5
3
4, 2:
4*x1^4*x2^2 + 8*x1^3*x2^3 + 4*x1^2*x2^4 + 8*x1^4*x2*x3 + 24*x1^3*x2^2*x3 + 24*x1^2*x2^3*x3 + 8*x1*x2^4*x3 + 4*x1^4*x3^2 + 24*x1^3*x2*x3^2 + 40*x1^2*x2^2*x3^2 + 24*x1*x2^3*x3^2 + 4*x2^4*x3^2 + 8*x1^3*x3^3 + 24*x1^2*x2*x3^3 + 24*x1*x2^2*x3^3 + 8*x2^3*x3^3 + 4*x1^2*x3^4 + 8*x1*x2*x3^4 + 4*x2^2*x3^4
3
3, 2, 1:
8*x1^3*x2^2*x3 + 8*x1^2*x2^3*x3 + 8*x1^3*x2*x3^2 + 16*x1^2*x2^2*x3^2 + 8*x1*x2^3*x3^2 + 8*x1^2*x2*x3^3 + 8*x1*x2^2*x3^3
4
2:
2*x1^2 + 4*x1*x2 + 2*x2^2 + 4*x1*x3 + 4*x2*x3 + 2*x3^2 + 4*x1*x4 + 4*x2*x4 + 4*x3*x4 + 2*x4^2
4
3:
2*x1^3 + 4*x1^2*x2 + 4*x1*x2^2 + 2*x2^3 + 4*x1^2*x3 + 8*x1*x2*x3 + 4*x2^2*x3 + 4*x1*x3^2 + 4*x2*x3^2 + 2*x3^3 + 4*x1^2*x4 + 8*x1*x2*x4 + 4*x2^2*x4 + 8*x1*x3*x4 + 8*x2*x3*x4 + 4*x3^2*x4 + 4*x1*x4^2 + 4*x2*x4^2 + 4*x3*x4^2 + 2*x4^3
4
2, 1:
4*x1^2*x2 + 4*x1*x2^2 + 4*x1^2*x3 + 8*x1*x2*x3 + 4*x2^2*x3 + 4*x1*x3^2 + 4*x2*x3^2 + 4*x1^2*x4 + 8*x1*x2*x4 + 4*x2^2*x4 + 8*x1*x3*x4 + 8*x2*x3*x4 + 4*x3^2*x4 + 4*x1*x4^2 + 4*x2*x4^2 + 4*x3*x4^2
4
4:
2*x1^4 + 4*x1^3*x2 + 4*x1^2*x2^2 + 4*x1*x2^3 + 2*x2^4 + 4*x1^3*x3 + 8*x1^2*x2*x3 + 8*x1*x2^2*x3 + 4*x2^3*x3 + 4*x1^2*x3^2 + 8*x1*x2*x3^2 + 4*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + 2*x3^4 + 4*x1^3*x4 + 8*x1^2*x2*x4 + 8*x1*x2^2*x4 + 4*x2^3*x4 + 8*x1^2*x3*x4 + 16*x1*x2*x3*x4 + 8*x2^2*x3*x4 + 8*x1*x3^2*x4 + 8*x2*x3^2*x4 + 4*x3^3*x4 + 4*x1^2*x4^2 + 8*x1*x2*x4^2 + 4*x2^2*x4^2 + 8*x1*x3*x4^2 + 8*x2*x3*x4^2 + 4*x3^2*x4^2 + 4*x1*x4^3 + 4*x2*x4^3 + 4*x3*x4^3 + 2*x4^4
4
3, 1:
4*x1^3*x2 + 8*x1^2*x2^2 + 4*x1*x2^3 + 4*x1^3*x3 + 16*x1^2*x2*x3 + 16*x1*x2^2*x3 + 4*x2^3*x3 + 8*x1^2*x3^2 + 16*x1*x2*x3^2 + 8*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + 4*x1^3*x4 + 16*x1^2*x2*x4 + 16*x1*x2^2*x4 + 4*x2^3*x4 + 16*x1^2*x3*x4 + 32*x1*x2*x3*x4 + 16*x2^2*x3*x4 + 16*x1*x3^2*x4 + 16*x2*x3^2*x4 + 4*x3^3*x4 + 8*x1^2*x4^2 + 16*x1*x2*x4^2 + 8*x2^2*x4^2 + 16*x1*x3*x4^2 + 16*x2*x3*x4^2 + 8*x3^2*x4^2 + 4*x1*x4^3 + 4*x2*x4^3 + 4*x3*x4^3
4
5:
2*x1^5 + 4*x1^4*x2 + 4*x1^3*x2^2 + 4*x1^2*x2^3 + 4*x1*x2^4 + 2*x2^5 + 4*x1^4*x3 + 8*x1^3*x2*x3 + 8*x1^2*x2^2*x3 + 8*x1*x2^3*x3 + 4*x2^4*x3 + 4*x1^3*x3^2 + 8*x1^2*x2*x3^2 + 8*x1*x2^2*x3^2 + 4*x2^3*x3^2 + 4*x1^2*x3^3 + 8*x1*x2*x3^3 + 4*x2^2*x3^3 + 4*x1*x3^4 + 4*x2*x3^4 + 2*x3^5 + 4*x1^4*x4 + 8*x1^3*x2*x4 + 8*x1^2*x2^2*x4 + 8*x1*x2^3*x4 + 4*x2^4*x4 + 8*x1^3*x3*x4 + 16*x1^2*x2*x3*x4 + 16*x1*x2^2*x3*x4 + 8*x2^3*x3*x4 + 8*x1^2*x3^2*x4 + 16*x1*x2*x3^2*x4 + 8*x2^2*x3^2*x4 + 8*x1*x3^3*x4 + 8*x2*x3^3*x4 + 4*x3^4*x4 + 4*x1^3*x4^2 + 8*x1^2*x2*x4^2 + 8*x1*x2^2*x4^2 + 4*x2^3*x4^2 + 8*x1^2*x3*x4^2 + 16*x1*x2*x3*x4^2 + 8*x2^2*x3*x4^2 + 8*x1*x3^2*x4^2 + 8*x2*x3^2*x4^2 + 4*x3^3*x4^2 + 4*x1^2*x4^3 + 8*x1*x2*x4^3 + 4*x2^2*x4^3 + 8*x1*x3*x4^3 + 8*x2*x3*x4^3 + 4*x3^2*x4^3 + 4*x1*x4^4 + 4*x2*x4^4 + 4*x3*x4^4 + 2*x4^5
4
4, 1:
4*x1^4*x2 + 8*x1^3*x2^2 + 8*x1^2*x2^3 + 4*x1*x2^4 + 4*x1^4*x3 + 16*x1^3*x2*x3 + 24*x1^2*x2^2*x3 + 16*x1*x2^3*x3 + 4*x2^4*x3 + 8*x1^3*x3^2 + 24*x1^2*x2*x3^2 + 24*x1*x2^2*x3^2 + 8*x2^3*x3^2 + 8*x1^2*x3^3 + 16*x1*x2*x3^3 + 8*x2^2*x3^3 + 4*x1*x3^4 + 4*x2*x3^4 + 4*x1^4*x4 + 16*x1^3*x2*x4 + 24*x1^2*x2^2*x4 + 16*x1*x2^3*x4 + 4*x2^4*x4 + 16*x1^3*x3*x4 + 48*x1^2*x2*x3*x4 + 48*x1*x2^2*x3*x4 + 16*x2^3*x3*x4 + 24*x1^2*x3^2*x4 + 48*x1*x2*x3^2*x4 + 24*x2^2*x3^2*x4 + 16*x1*x3^3*x4 + 16*x2*x3^3*x4 + 4*x3^4*x4 + 8*x1^3*x4^2 + 24*x1^2*x2*x4^2 + 24*x1*x2^2*x4^2 + 8*x2^3*x4^2 + 24*x1^2*x3*x4^2 + 48*x1*x2*x3*x4^2 + 24*x2^2*x3*x4^2 + 24*x1*x3^2*x4^2 + 24*x2*x3^2*x4^2 + 8*x3^3*x4^2 + 8*x1^2*x4^3 + 16*x1*x2*x4^3 + 8*x2^2*x4^3 + 16*x1*x3*x4^3 + 16*x2*x3*x4^3 + 8*x3^2*x4^3 + 4*x1*x4^4 + 4*x2*x4^4 + 4*x3*x4^4
4
3, 2:
4*x1^3*x2^2 + 4*x1^2*x2^3 + 8*x1^3*x2*x3 + 16*x1^2*x2^2*x3 + 8*x1*x2^3*x3 + 4*x1^3*x3^2 + 16*x1^2*x2*x3^2 + 16*x1*x2^2*x3^2 + 4*x2^3*x3^2 + 4*x1^2*x3^3 + 8*x1*x2*x3^3 + 4*x2^2*x3^3 + 8*x1^3*x2*x4 + 16*x1^2*x2^2*x4 + 8*x1*x2^3*x4 + 8*x1^3*x3*x4 + 32*x1^2*x2*x3*x4 + 32*x1*x2^2*x3*x4 + 8*x2^3*x3*x4 + 16*x1^2*x3^2*x4 + 32*x1*x2*x3^2*x4 + 16*x2^2*x3^2*x4 + 8*x1*x3^3*x4 + 8*x2*x3^3*x4 + 4*x1^3*x4^2 + 16*x1^2*x2*x4^2 + 16*x1*x2^2*x4^2 + 4*x2^3*x4^2 + 16*x1^2*x3*x4^2 + 32*x1*x2*x3*x4^2 + 16*x2^2*x3*x4^2 + 16*x1*x3^2*x4^2 + 16*x2*x3^2*x4^2 + 4*x3^3*x4^2 + 4*x1^2*x4^3 + 8*x1*x2*x4^3 + 4*x2^2*x4^3 + 8*x1*x3*x4^3 + 8*x2*x3*x4^3 + 4*x3^2*x4^3
4
6:
2*x1^6 + 4*x1^5*x2 + 4*x1^4*x2^2 + 4*x1^3*x2^3 + 4*x1^2*x2^4 + 4*x1*x2^5 + 2*x2^6 + 4*x1^5*x3 + 8*x1^4*x2*x3 + 8*x1^3*x2^2*x3 + 8*x1^2*x2^3*x3 + 8*x1*x2^4*x3 + 4*x2^5*x3 + 4*x1^4*x3^2 + 8*x1^3*x2*x3^2 + 8*x1^2*x2^2*x3^2 + 8*x1*x2^3*x3^2 + 4*x2^4*x3^2 + 4*x1^3*x3^3 + 8*x1^2*x2*x3^3 + 8*x1*x2^2*x3^3 + 4*x2^3*x3^3 + 4*x1^2*x3^4 + 8*x1*x2*x3^4 + 4*x2^2*x3^4 + 4*x1*x3^5 + 4*x2*x3^5 + 2*x3^6 + 4*x1^5*x4 + 8*x1^4*x2*x4 + 8*x1^3*x2^2*x4 + 8*x1^2*x2^3*x4 + 8*x1*x2^4*x4 + 4*x2^5*x4 + 8*x1^4*x3*x4 + 16*x1^3*x2*x3*x4 + 16*x1^2*x2^2*x3*x4 + 16*x1*x2^3*x3*x4 + 8*x2^4*x3*x4 + 8*x1^3*x3^2*x4 + 16*x1^2*x2*x3^2*x4 + 16*x1*x2^2*x3^2*x4 + 8*x2^3*x3^2*x4 + 8*x1^2*x3^3*x4 + 16*x1*x2*x3^3*x4 + 8*x2^2*x3^3*x4 + 8*x1*x3^4*x4 + 8*x2*x3^4*x4 + 4*x3^5*x4 + 4*x1^4*x4^2 + 8*x1^3*x2*x4^2 + 8*x1^2*x2^2*x4^2 + 8*x1*x2^3*x4^2 + 4*x2^4*x4^2 + 8*x1^3*x3*x4^2 + 16*x1^2*x2*x3*x4^2 + 16*x1*x2^2*x3*x4^2 + 8*x2^3*x3*x4^2 + 8*x1^2*x3^2*x4^2 + 16*x1*x2*x3^2*x4^2 + 8*x2^2*x3^2*x4^2 + 8*x1*x3^3*x4^2 + 8*x2*x3^3*x4^2 + 4*x3^4*x4^2 + 4*x1^3*x4^3 + 8*x1^2*x2*x4^3 + 8*x1*x2^2*x4^3 + 4*x2^3*x4^3 + 8*x1^2*x3*x4^3 + 16*x1*x2*x3*x4^3 + 8*x2^2*x3*x4^3 + 8*x1*x3^2*x4^3 + 8*x2*x3^2*x4^3 + 4*x3^3*x4^3 + 4*x1^2*x4^4 + 8*x1*x2*x4^4 + 4*x2^2*x4^4 + 8*x1*x3*x4^4 + 8*x2*x3*x4^4 + 4*x3^2*x4^4 + 4*x1*x4^5 + 4*x2*x4^5 + 4*x3*x4^5 + 2*x4^6
4
5, 1:
4*x1^5*x2 + 8*x1^4*x2^2 + 8*x1^3*x2^3 + 8*x1^2*x2^4 + 4*x1*x2^5 + 4*x1^5*x3 + 16*x1^4*x2*x3 + 24*x1^3*x2^2*x3 + 24*x1^2*x2^3*x3 + 16*x1*x2^4*x3 + 4*x2^5*x3 + 8*x1^4*x3^2 + 24*x1^3*x2*x3^2 + 32*x1^2*x2^2*x3^2 + 24*x1*x2^3*x3^2 + 8*x2^4*x3^2 + 8*x1^3*x3^3 + 24*x1^2*x2*x3^3 + 24*x1*x2^2*x3^3 + 8*x2^3*x3^3 + 8*x1^2*x3^4 + 16*x1*x2*x3^4 + 8*x2^2*x3^4 + 4*x1*x3^5 + 4*x2*x3^5 + 4*x1^5*x4 + 16*x1^4*x2*x4 + 24*x1^3*x2^2*x4 + 24*x1^2*x2^3*x4 + 16*x1*x2^4*x4 + 4*x2^5*x4 + 16*x1^4*x3*x4 + 48*x1^3*x2*x3*x4 + 64*x1^2*x2^2*x3*x4 + 48*x1*x2^3*x3*x4 + 16*x2^4*x3*x4 + 24*x1^3*x3^2*x4 + 64*x1^2*x2*x3^2*x4 + 64*x1*x2^2*x3^2*x4 + 24*x2^3*x3^2*x4 + 24*x1^2*x3^3*x4 + 48*x1*x2*x3^3*x4 + 24*x2^2*x3^3*x4 + 16*x1*x3^4*x4 + 16*x2*x3^4*x4 + 4*x3^5*x4 + 8*x1^4*x4^2 + 24*x1^3*x2*x4^2 + 32*x1^2*x2^2*x4^2 + 24*x1*x2^3*x4^2 + 8*x2^4*x4^2 + 24*x1^3*x3*x4^2 + 64*x1^2*x2*x3*x4^2 + 64*x1*x2^2*x3*x4^2 + 24*x2^3*x3*x4^2 + 32*x1^2*x3^2*x4^2 + 64*x1*x2*x3^2*x4^2 + 32*x2^2*x3^2*x4^2 + 24*x1*x3^3*x4^2 + 24*x2*x3^3*x4^2 + 8*x3^4*x4^2 + 8*x1^3*x4^3 + 24*x1^2*x2*x4^3 + 24*x1*x2^2*x4^3 + 8*x2^3*x4^3 + 24*x1^2*x3*x4^3 + 48*x1*x2*x3*x4^3 + 24*x2^2*x3*x4^3 + 24*x1*x3^2*x4^3 + 24*x2*x3^2*x4^3 + 8*x3^3*x4^3 + 8*x1^2*x4^4 + 16*x1*x2*x4^4 + 8*x2^2*x4^4 + 16*x1*x3*x4^4 + 16*x2*x3*x4^4 + 8*x3^2*x4^4 + 4*x1*x4^5 + 4*x2*x4^5 + 4*x3*x4^5
4
4, 2:
4*x1^4*x2^2 + 8*x1^3*x2^3 + 4*x1^2*x2^4 + 8*x1^4*x2*x3 + 24*x1^3*x2^2*x3 + 24*x1^2*x2^3*x3 + 8*x1*x2^4*x3 + 4*x1^4*x3^2 + 24*x1^3*x2*x3^2 + 40*x1^2*x2^2*x3^2 + 24*x1*x2^3*x3^2 + 4*x2^4*x3^2 + 8*x1^3*x3^3 + 24*x1^2*x2*x3^3 + 24*x1*x2^2*x3^3 + 8*x2^3*x3^3 + 4*x1^2*x3^4 + 8*x1*x2*x3^4 + 4*x2^2*x3^4 + 8*x1^4*x2*x4 + 24*x1^3*x2^2*x4 + 24*x1^2*x2^3*x4 + 8*x1*x2^4*x4 + 8*x1^4*x3*x4 + 48*x1^3*x2*x3*x4 + 80*x1^2*x2^2*x3*x4 + 48*x1*x2^3*x3*x4 + 8*x2^4*x3*x4 + 24*x1^3*x3^2*x4 + 80*x1^2*x2*x3^2*x4 + 80*x1*x2^2*x3^2*x4 + 24*x2^3*x3^2*x4 + 24*x1^2*x3^3*x4 + 48*x1*x2*x3^3*x4 + 24*x2^2*x3^3*x4 + 8*x1*x3^4*x4 + 8*x2*x3^4*x4 + 4*x1^4*x4^2 + 24*x1^3*x2*x4^2 + 40*x1^2*x2^2*x4^2 + 24*x1*x2^3*x4^2 + 4*x2^4*x4^2 + 24*x1^3*x3*x4^2 + 80*x1^2*x2*x3*x4^2 + 80*x1*x2^2*x3*x4^2 + 24*x2^3*x3*x4^2 + 40*x1^2*x3^2*x4^2 + 80*x1*x2*x3^2*x4^2 + 40*x2^2*x3^2*x4^2 + 24*x1*x3^3*x4^2 + 24*x2*x3^3*x4^2 + 4*x3^4*x4^2 + 8*x1^3*x4^3 + 24*x1^2*x2*x4^3 + 24*x1*x2^2*x4^3 + 8*x2^3*x4^3 + 24*x1^2*x3*x4^3 + 48*x1*x2*x3*x4^3 + 24*x2^2*x3*x4^3 + 24*x1*x3^2*x4^3 + 24*x2*x3^2*x4^3 + 8*x3^3*x4^3 + 4*x1^2*x4^4 + 8*x1*x2*x4^4 + 4*x2^2*x4^4 + 8*x1*x3*x4^4 + 8*x2*x3*x4^4 + 4*x3^2*x4^4
4
3, 2, 1:
8*x1^3*x2^2*x3 + 8*x1^2*x2^3*x3 + 8*x1^3*x2*x3^2 + 16*x1^2*x2^2*x3^2 + 8*x1*x2^3*x3^2 + 8*x1^2*x2*x3^3 + 8*x1*x2^2*x3^3 + 8*x1^3*x2^2*x4 + 8*x1^2*x2^3*x4 + 16*x1^3*x2*x3*x4 + 32*x1^2*x2^2*x3*x4 + 16*x1*x2^3*x3*x4 + 8*x1^3*x3^2*x4 + 32*x1^2*x2*x3^2*x4 + 32*x1*x2^2*x3^2*x4 + 8*x2^3*x3^2*x4 + 8*x1^2*x3^3*x4 + 16*x1*x2*x3^3*x4 + 8*x2^2*x3^3*x4 + 8*x1^3*x2*x4^2 + 16*x1^2*x2^2*x4^2 + 8*x1*x2^3*x4^2 + 8*x1^3*x3*x4^2 + 32*x1^2*x2*x3*x4^2 + 32*x1*x2^2*x3*x4^2 + 8*x2^3*x3*x4^2 + 16*x1^2*x3^2*x4^2 + 32*x1*x2*x3^2*x4^2 + 16*x2^2*x3^2*x4^2 + 8*x1*x3^3*x4^2 + 8*x2*x3^3*x4^2 + 8*x1^2*x2*x4^3 + 8*x1*x2^2*x4^3 + 8*x1^2*x3*x4^3 + 16*x1*x2*x3*x4^3 + 8*x2^2*x3*x4^3 + 8*x1*x3^2*x4^3 + 8*x2*x3^2*x4^3
5
2:
2*x1^2 + 4*x1*x2 + 2*x2^2 + 4*x1*x3 + 4*x2*x3 + 2*x3^2 + 4*x1*x4 + 4*x2*x4 + 4*x3*x4 + 2*x4^2 + 4*x1*x5 + 4*x2*x5 + 4*x3*x5 + 4*x4*x5 + 2*x5^2
5
3:
2*x1^3 + 4*x1^2*x2 + 4*x1*x2^2 + 2*x2^3 + 4*x1^2*x3 + 8*x1*x2*x3 + 4*x2^2*x3 + 4*x1*x3^2 + 4*x2*x3^2 + 2*x3^3 + 4*x1^2*x4 + 8*x1*x2*x4 + 4*x2^2*x4 + 8*x1*x3*x4 + 8*x2*x3*x4 + 4*x3^2*x4 + 4*x1*x4^2 + 4*x2*x4^2 + 4*x3*x4^2 + 2*x4^3 + 4*x1^2*x5 + 8*x1*x2*x5 + 4*x2^2*x5 + 8*x1*x3*x5 + 8*x2*x3*x5 + 4*x3^2*x5 + 8*x1*x4*x5 + 8*x2*x4*x5 + 8*x3*x4*x5 + 4*x4^2*x5 + 4*x1*x5^2 + 4*x2*x5^2 + 4*x3*x5^2 + 4*x4*x5^2 + 2*x5^3
5
2, 1:
4*x1^2*x2 + 4*x1*x2^2 + 4*x1^2*x3 + 8*x1*x2*x3 + 4*x2^2*x3 + 4*x1*x3^2 + 4*x2*x3^2 + 4*x1^2*x4 + 8*x1*x2*x4 + 4*x2^2*x4 + 8*x1*x3*x4 + 8*x2*x3*x4 + 4*x3^2*x4 + 4*x1*x4^2 + 4*x2*x4^2 + 4*x3*x4^2 + 4*x1^2*x5 + 8*x1*x2*x5 + 4*x2^2*x5 + 8*x1*x3*x5 + 8*x2*x3*x5 + 4*x3^2*x5 + 8*x1*x4*x5 + 8*x2*x4*x5 + 8*x3*x4*x5 + 4*x4^2*x5 + 4*x1*x5^2 + 4*x2*x5^2 + 4*x3*x5^2 + 4*x4*x5^2
5
4:
2*x1^4 + 4*x1^3*x2 + 4*x1^2*x2^2 + 4*x1*x2^3 + 2*x2^4 + 4*x1^3*x3 + 8*x1^2*x2*x3 + 8*x1*x2^2*x3 + 4*x2^3*x3 + 4*x1^2*x3^2 + 8*x1*x2*x3^2 + 4*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + 2*x3^4 + 4*x1^3*x4 + 8*x1^2*x2*x4 + 8*x1*x2^2*x4 + 4*x2^3*x4 + 8*x1^2*x3*x4 + 16*x1*x2*x3*x4 + 8*x2^2*x3*x4 + 8*x1*x3^2*x4 + 8*x2*x3^2*x4 + 4*x3^3*x4 + 4*x1^2*x4^2 + 8*x1*x2*x4^2 + 4*x2^2*x4^2 + 8*x1*x3*x4^2 + 8*x2*x3*x4^2 + 4*x3^2*x4^2 + 4*x1*x4^3 + 4*x2*x4^3 + 4*x3*x4^3 + 2*x4^4 + 4*x1^3*x5 + 8*x1^2*x2*x5 + 8*x1*x2^2*x5 + 4*x2^3*x5 + 8*x1^2*x3*x5 + 16*x1*x2*x3*x5 + 8*x2^2*x3*x5 + 8*x1*x3^2*x5 + 8*x2*x3^2*x5 + 4*x3^3*x5 + 8*x1^2*x4*x5 + 16*x1*x2*x4*x5 + 8*x2^2*x4*x5 + 16*x1*x3*x4*x5 + 16*x2*x3*x4*x5 + 8*x3^2*x4*x5 + 8*x1*x4^2*x5 + 8*x2*x4^2*x5 + 8*x3*x4^2*x5 + 4*x4^3*x5 + 4*x1^2*x5^2 + 8*x1*x2*x5^2 + 4*x2^2*x5^2 + 8*x1*x3*x5^2 + 8*x2*x3*x5^2 + 4*x3^2*x5^2 + 8*x1*x4*x5^2 + 8*x2*x4*x5^2 + 8*x3*x4*x5^2 + 4*x4^2*x5^2 + 4*x1*x5^3 + 4*x2*x5^3 + 4*x3*x5^3 + 4*x4*x5^3 + 2*x5^4
5
3, 1:
4*x1^3*x2 + 8*x1^2*x2^2 + 4*x1*x2^3 + 4*x1^3*x3 + 16*x1^2*x2*x3 + 16*x1*x2^2*x3 + 4*x2^3*x3 + 8*x1^2*x3^2 + 16*x1*x2*x3^2 + 8*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + 4*x1^3*x4 + 16*x1^2*x2*x4 + 16*x1*x2^2*x4 + 4*x2^3*x4 + 16*x1^2*x3*x4 + 32*x1*x2*x3*x4 + 16*x2^2*x3*x4 + 16*x1*x3^2*x4 + 16*x2*x3^2*x4 + 4*x3^3*x4 + 8*x1^2*x4^2 + 16*x1*x2*x4^2 + 8*x2^2*x4^2 + 16*x1*x3*x4^2 + 16*x2*x3*x4^2 + 8*x3^2*x4^2 + 4*x1*x4^3 + 4*x2*x4^3 + 4*x3*x4^3 + 4*x1^3*x5 + 16*x1^2*x2*x5 + 16*x1*x2^2*x5 + 4*x2^3*x5 + 16*x1^2*x3*x5 + 32*x1*x2*x3*x5 + 16*x2^2*x3*x5 + 16*x1*x3^2*x5 + 16*x2*x3^2*x5 + 4*x3^3*x5 + 16*x1^2*x4*x5 + 32*x1*x2*x4*x5 + 16*x2^2*x4*x5 + 32*x1*x3*x4*x5 + 32*x2*x3*x4*x5 + 16*x3^2*x4*x5 + 16*x1*x4^2*x5 + 16*x2*x4^2*x5 + 16*x3*x4^2*x5 + 4*x4^3*x5 + 8*x1^2*x5^2 + 16*x1*x2*x5^2 + 8*x2^2*x5^2 + 16*x1*x3*x5^2 + 16*x2*x3*x5^2 + 8*x3^2*x5^2 + 16*x1*x4*x5^2 + 16*x2*x4*x5^2 + 16*x3*x4*x5^2 + 8*x4^2*x5^2 + 4*x1*x5^3 + 4*x2*x5^3 + 4*x3*x5^3 + 4*x4*x5^3
Definition
The Schur Q-polynomial $Q_\lambda(x_1,\ldots,x_n)$ is the symmetric polynomial defined by the one-row generating function (1) and the Pfaffian (2), with the standard $Q$ normalization [1]. It is the specialization of the Schur Q-function to the alphabet $x_1,\ldots,x_n$.
Parameters
$n$
—   number of alphabet variables (a positive integer)
$\lambda$
—   strict partition (a strictly decreasing sequence of positive integers with length at most $n$)
Formulas
(1)
$\displaystyle\sum_{r\geq0}q_r(x_1,\ldots,x_n)z^r=\prod_{i=1}^n\frac{1+x_i z}{1-x_i z}$, with $q_0=1$ and $Q_{(r)}=q_r$ for $r>0$ [1].
(2)
For $a>b\geq0$, put $Q_{(a,b)}=q_aq_b+2\sum_{j=1}^b(-1)^j q_{a+j}q_{b-j}$. Append a single zero to $\lambda$ when $\ell(\lambda)$ is odd, obtaining an even-length sequence $\mu$. Define the skew-symmetric matrix $A_\lambda$ by $(A_\lambda)_{ij}=Q_{(\mu_i,\mu_j)}$ for $i<j$ and zero diagonal. Then $Q_\lambda=\operatorname{Pf}(A_\lambda)$ [1].
(3)
$\displaystyle Q_\lambda(x_1,\ldots,x_n)=\sum_T\prod_{k=1}^n x_k^{a_k(T)}$, where $T$ runs over the marked shifted tableaux specified in (9), and $a_k(T)$ counts entries equal to $k$ or $k^\prime$ [2].
(4)
$Q_\lambda(x_1,\ldots,x_n)=2^{\ell(\lambda)}P^{\mathrm{HL}}_\lambda(x_1,\ldots,x_n;-1)$, where $P^{\mathrm{HL}}_\lambda$ is the Hall–Littlewood polynomial in the $P$ normalization [3].
(5)
For $\delta_m=(m,m-1,\ldots,1)$, $Q_{\delta_m}(x)=2^m s_{\delta_m}(x)$, where $s_\lambda$ is the Schur polynomial [2].
(6)
$Q_\lambda(x_1,\ldots,x_{n-1},0)=Q_\lambda(x_1,\ldots,x_{n-1})$ [1]. The right side is zero when $\ell(\lambda)>n-1$.
(7)
$Q_\varnothing=1$, $Q_{(1)}(x_1,\ldots,x_n)=2\sum_{i=1}^n x_i$, and $Q_{(d)}(x_1)=2x_1^d$ for $d\geq1$.
Comments
(8)
A strict partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ has $\lambda_1>\cdots>\lambda_\ell>0$, length $\ell=\ell(\lambda)$ and weight $d=|\lambda|=\sum_i\lambda_i$. Entry identities list its parts separated by commas, without zero padding. The variables $x_1,\ldots,x_n$ are formal indeterminates; each stored polynomial is fully expanded over $\mathbb Z$.
(9)
The shifted diagram of $\lambda$ has cells $(i,j)$ with $1\leq i\leq\ell$ and $i\leq j\leq i+\lambda_i-1$. Fill it from the ordered alphabet $1^\prime<1<\cdots<n^\prime<n$, weakly increasing along rows and down columns, with at most one occurrence of each primed symbol per row and each unprimed symbol per column. Primed entries are allowed on the diagonal [2].
(10)
Before specializing the alphabet, the Schur Q-functions lie in the $\mathbb Q$-subalgebra generated by the odd power sums $p_1,p_3,p_5,\ldots$, where $p_r=\sum_i x_i^r$ [3]. The Schur $P$ convention is $P_\lambda=2^{-\ell(\lambda)}Q_\lambda$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
import sage.rings.polynomial.laurent_polynomial_ring
from sage.symbolic.ring import SR
from sage.combinat.sf.sf import SymmetricFunctions

# One entry, in the Schur Q normalization.
n, lam = 3, [3, 2]
Q = SymmetricFunctions(QQ).hall_littlewood(t=-1).Q()
R = PolynomialRing(ZZ, n, names=['x%d' % i for i in range(1, n+1)])
print(R(str(Q[lam].expand(n, alphabet=list(R.variable_names())))))
References
[1]
S. Okada, Pfaffian Formulas and Schur Q-Function Identities, Advances in Mathematics 353 (2019), 446–470, Theorem 4.1 and Lemma 4.2. (arXiv) (doi)
[2]
H. Rosengren, Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux, Journal of Combinatorial Theory, Series A 115 (2008), 376–406, Section 2.1, equations (2.1)–(2.3), and the staircase case in the introduction. (arXiv) (doi)
Links
Similar tables
Hall–Littlewood polynomials —   give $Q_\lambda$ at $t=-1$ after multiplying the $P$ normalization by $2^{\ell(\lambda)}$
Schur polynomials —   agree with $2^{-m}Q_{(m,m-1,\ldots,1)}$ on staircase shapes
power sum symmetric polynomials —   generate over $\mathbb Q$ the Schur Q subalgebra through the odd-indexed power sums
Data properties
Entries are of type: integral polynomial
How well the digits are known: exact
Table is complete: no (it holds every strict partition with $\ell(\lambda)\leq n$ and $2\leq|\lambda|\leq6$ for $n=2,3,4$, and $2\leq|\lambda|\leq4$ for $n=5$, giving 40 entries. These complete degree slices cover one-row and multirow shapes across small alphabets, including the first three-row shape $(3,2,1)$. The noncanonical bounds are chosen for readability: the longest entry has 1299 characters, while the omitted five-variable degree-5 slice reaches 1937. Degree zero, degree one and the one-variable alphabet are covered by (7). The attached argument-selection.md explains both parameter families and the finite bounds)
How they were obtained:

All coefficients are exact integers. The generator builds the one-row generating function coefficient by coefficient in $\mathbb Z[x_1,\ldots,x_n]$, then expands the Pfaffian recursively using only addition and multiplication.

more

Every polynomial is compared coefficient by coefficient with a separate enumeration of marked shifted tableaux [2] and with SageMath's Hall–Littlewood $Q$ basis at $t=-1$ [3]. Known linear, quadratic and $(2,1)$ expansions, including an impossible one-variable shape, serve as controls. Checks also cover enumeration, symmetry, homogeneity, coefficient divisibility, alphabet stability, opposite-pair cancellation, odd power-sum support, boundary formulas and staircase Schur identities. Fifteen overlapping entries are compared with the stored Hall–Littlewood table, and three staircase entries with the stored Schur table. The same checks are applied to values read back from the draft.



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