A090349 Pascal-like triangle read by rows. Number of nonterminal symbols (which generate strings of length k) in a certain "divide-and-conquer" context-free grammar in Chomsky normal form that generates all permutations of n symbols.

1, 2, 1, 3, 3, 1, 4, 6, 0, 1, 5, 10, 10, 0, 1, 6, 15, 20, 0, 0, 1, 7, 21, 35, 35, 0, 0, 1, 8, 28, 0, 70, 0, 0, 0, 1, 9, 36, 84, 126, 126, 0, 0, 0, 1, 10, 45, 120, 0, 252, 0, 0, 0, 0, 1

Offset

1,2

Links

Table of n, a(n) for n=1..55.

P. R. J. Asveld, Generating all permutations by context-free grammars in Chomsky normal form, Theoretical Computer Science 354 (2006) 118-130.

Formula

a(n,k) = C(n,k), if k = ceiling(n/(2^i)) or k = floor(n/(2^i)) for some i with 0 <= i <= ceiling(log_2 n); a(n,k)=0 otherwise.

Example

Triangle begins:
   1;
   2,  1;
   3,  3,  1;
   4,  6,  0,    1;
   5, 10,  10,   0,  1;
   6, 15,  20,   0,  0,  1;
   7, 21,  35,  35,  0,  0, 1;
   8, 28,   0,  70,  0,  0, 0, 1;
   9, 36,  84, 126, 126, 0, 0, 0, 1;
  10, 45, 120,   0, 252, 0, 0, 0, 0, 1;
  ...
Example grammar: S -> EF | FE | GH | HG | IJ | JI, E -> AB | BA, F -> CD | DC, G -> AC | CA, H -> BD | DB, I -> AD | DA, J-> BC | CB, A-> a, B-> b, C-> c, D -> d; so a(4,4)=#{S}=1, a(4,3)=#{}=0, a(4,2)=#{E,F,G,H,I,J}=6 and a(4,1)= #{A,B,C,D}=4.

Crossrefs

Sequence in context: A185095 A177888 A073020 * A157379 A212139 A093430

Adjacent sequences: A090346 A090347 A090348 * A090350 A090351 A090352

Keyword

nonn,tabl

Author

Peter R. J. Asveld, Jan 29 2004

Status

approved