The 𝑆 𝑛 -equivariant Euler characteristic of ℳ 1,𝑛 (ℙ 𝑟 , 𝑑)

We compute the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> Sₙ -equivariant topological Euler characteristic of the Kontsevich moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi mathvariant="script">M</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> M̄1,n(Pʳ,d) . Letting <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mover accent="true"> <m:mi mathvariant="script">M</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mi>nrt</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊂</m:mo> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi mathvariant="script">M</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> M̄1,nⁿʳᵗ(Pʳ,d)̄1,n(Pʳ,d) denote the subspace of maps from curves without rational tails, we solve for the motive of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi mathvariant="script">M</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> M̄1,n(Pʳ,d) in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mover accent="true"> <m:mi mathvariant="script">M</m:mi> <m:mo mathvariant="italic">̄</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mi>nrt</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> M̄1,nⁿʳᵗ(Pʳ,d) and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mo>⋆</m:mo> </m:msup> </m:math> C^{}