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COMMENTS
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a(n)=Sum[f(L)^2 Sum h(v)^2], where L is a partition of n, f(L) is the number of standard Young tableaux of shape L, v is a box in L (i.e. in the Ferrers diagram of L), h(v) is the hook length of v, the inner summation is over all boxes v in L and the outer summation is over all partitions of n. Example:
a(3)=72 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the values of f(L) are 1, 2, 1, respectively, the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, Sum h(v)^2 = 14, 11, 14, respectively and now a(n)=1^2*14+2^2*11+1^2*14=72.
Number of marked permutations of [n], i.e. permutations of [n] where the entry j (1<=j<=n) can be marked by any integer k satisfying 1<=k<=n+j-1. Example: a(2)=10 because we have (the mark k is placed between parentheses following the marked entry j): 1(1)2, 1(2)2, 21(1),21(2),12(1),12(2),12(3),2(1)1,2(2)1,2(3)1.
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REFERENCES
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Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (pp. 5, 28).
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398v1 [math.CO] 9 May 2008 (p. 4).
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