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For any two subsets $A$ and $B$ of $[n]$, we define the elements \\[ \\nabla_{B,A}:=\\sum_{\\substack{w\\in S_n;\\\\w\\left( A\\right) =B}} w \\qquad \\text{and} \\qquad \\widetilde{\\nabla}_{B,A}:=\\sum_{\\substack{w\\in S_n;\\\\w\\left( A\\right) \\subseteq B}}w \\] of $\\mathcal{A}$. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). 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For any two subsets $A$ and $B$ of $[n]$, we define the elements \\[ \\nabla_{B,A}:=\\sum_{\\substack{w\\in S_n;\\\\w\\left( A\\right) =B}} w \\qquad \\text{and} \\qquad \\widetilde{\\nabla}_{B,A}:=\\sum_{\\substack{w\\in S_n;\\\\w\\left( A\\right) \\subseteq B}}w \\] of $\\mathcal{A}$. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). 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