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SUMMARY:Gwenaël Richomme
DTSTART:20260317T140000Z
DTEND:20260317T150000Z
DTSTAMP:20260427T053621Z
UID:CombinatoricsOnWords/132
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Combinatoric
 sOnWords/132/">On some 2-binomial coefficients of binary words: geometrica
 l interpretation\, partitions of integers\, and fair words</a>\nby Gwenaë
 l Richomme as part of One World Combinatorics on Words Seminar\n\n\nAbstra
 ct\nThe binomial notation $\\binom{w}{u}$ represents the number of occurre
 nces of the word\n$u$ as a (scattered) subword in $w$. We first introduce 
 and study possible uses of\na geometrical interpretation of $\\binom{w}{ab
 }$ and \\binom{w}{ba} when $a$ and $b$ are distinct\nletters. We then stud
 y the structure of the $2$-binomial equivalence class of a\nbinary word $w
 $ (two words are $2$-binomially equivalent if they have the same\nbinomial
  coefficients\, that is\, the same numbers of occurrences\, for each word\
 nof length at most $2$). Especially we explain the existence of an isomorp
 hism\nbetween the graph of the $2$-binomial equivalence class of $w$ with 
 respect to a\nparticular rewriting rule and the lattice of partitions of t
 he integer \\binom{w}{ab}\nwith \\binom{w}{a} parts and greatest part boun
 ded by \\binom{w}{b}. Finally we study binary\nfair words\, the words over
  $\\{a\, b\\}$ having the same numbers of occurrences of $ab$\nand $ba$ as
  subwords $(\\binom{w}{ab} = \\binom{w}{ba})$. In particular\, we sketch a
  proof of a recent\nconjecture related to a special case of the least squa
 re approximation.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsOnWords/132/
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