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SUMMARY:Dennis Eichhorn (University of California - Irvine)
DTSTART:20260717T140000Z
DTEND:20260717T142500Z
DTSTAMP:20260710T111540Z
UID:CANT2026/61
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/61/
 ">The combinatorics of sequences that enjoy a curious self-convolutive pro
 perty</a>\nby Dennis Eichhorn (University of California - Irvine) as part 
 of Combinatorial and additive number theory seminar (CANT 2026)\n\nLecture
  held in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstra
 ct\nIn 2002\, Andrews\, Lewis\, and Lovejoy introduced the combinatorial o
 bjects called partitions with designated summands.\nIf we restrict our att
 ention to $\\mathrm{PDO}(n)$\, the number of partitions with designated su
 mmands in which all parts are odd\, a very curious property emerges.\nThe 
 very unexpected identity $\\qquad\n \\sum_{n=0}^\\infty \\mathrm{PDO}(2n)q
 ^n = \\left ( \\sum_{n=0}^\\infty \\mathrm{PDO}(n)q^n \\right )^2\n$ holds
 .\nThat is\, the sequence $\\{\\mathrm{PDO}(2n)\\}_{n=0}^\\infty$ is the c
 onvolution of the sequence $\\{\\mathrm{PDO}(n) \\}_{n=0}^\\infty$ with it
 self!\nSequences sharing this curious property are now called ``$2$-convol
 utive\,'' and a small handful of such sequences appear in the OEIS. Many a
 uthors have called for a combinatorial proof of the $2$-convolutivity of $
 \\mathrm{PDO}(n)$. After a nearly two-year-long collaboration with Chern\,
  Fu\, and Sellers\, we are happy to announce that we have finally found th
 e requested combinatorial proof.\nIn this talk\, we discuss this new proof
 \, along with the combinatorial proofs of the $2$-convolutivity of several
  other partition functions.\n
LOCATION:https://researchseminars.org/talk/CANT2026/61/
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