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3-dimensional generalized Catalan numbers
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3-dimensional generalized Catalan numbers in sequence first appeared in work of A. Reznik, V. Efimov, A. Solov’ev, and A. Torgov from the United Institute of Automation and Electrometry (Novosibirsk State University). Example of usage Various words of length (l+m+n) are composed from l symbols “a”, m symbols “b”, and n symbols “c”. Determine how many words are arranged in such a manner that the number of symbols “b” never exceeds the number of symbols “a” if the words are read from left to right and the number of symbols “c” never exceeds the number of symbols “a” if the words are read from right to left. <math> Q(l,m,n) = ((l+m+n)!/(l!m!n!)) * ((l+1)(l+2)-(m+n)(l+2)+m*n)/(l+1)(l+2) </math> |
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