The stable defective Kreweras number formula

From papers

Let m,d,nm,d,n be nonnegative integers, and let 4λm4\lambda\vdash m be a partition of mm of length kk. For each ii, let 4μi(λ)4\mu_i(\lambda) be the number of parts of 4λ4\lambda of size ii. For nd+m1n\geq d+m-1, the defective Kreweras number conjecture.

Krewd,n(λ)=m+dkm+d1m+d+1(m+d+1m+d+1k,μ1(λ),μ2(λ),,μm(λ)).\mathrm{Krew}_{d,n}(\lambda)=\frac{m+dk}{m+d}\frac{1}{m+d+1}{m+d+1\choose m+d+1-k,\mu_1(\lambda),\mu_2(\lambda),\ldots,\mu_m(\lambda)}.

This conjectured formula is a stable-range analogue of the formula for the ordinary Kreweras numbers. The supplied text gives no evidence that the conjecture has been proved or disproved, so its status remains open.

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Sources & referencesView supporting material

Primary source

Rebecca E. Garcia, Pamela E. Harris, Alex Moon, Aaron Ortiz, Lauren J. Quesada, Cynthia Marie Rivera Sánchez, Dwight Anderson Williams and Alexander N. Wilson, “The Defective Parking Space and Defective Kreweras Numbers”, arXiv:2405.14635 (2026).

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