Hafnian is a special determinant-like expression evaluated for the pair $(F,I)$ of a matrix corresponding to an element $F$ of symplectic Lie algebra $sp(2n)$ and a subsequence $\mathbf{i} = (i_1,\ldots,i_{2k})$ of the ascending sequence $(-n,\ldots,-1, 1,\ldots,n)$. Let $\Sigma_n$ be the symmetric group on $n$ letters. Then
Related entries include determinant, Pfaffian, Pfaffian line bundle
J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026
sec 4.4. in: A. I. Molev, Yangians and their applications, in “Handbook of Algebra”, Vol. 3, (M. Hazewinkel, Ed.), Elsevier, 2003, pp. 907-959 http://arxiv.org/abs/math.QA/0211288