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Search: id:A135313
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| A135313 |
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Triangle of numbers T(n,k) (n>=0, n>=k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where k=max_{x}(|{y : xRy}|), read by rows. Early confluency means that (xRy AND xRz) implies (yRz OR zRy) for all x, y, z. |
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3
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| 1, 0, 1, 0, 1, 3, 0, 1, 12, 13, 0, 1, 61, 106, 75, 0, 1, 310, 1105, 1035, 541, 0, 1, 1821, 12075, 16025, 11301, 4683, 0, 1, 11592, 141533, 267715, 239379, 137774, 47293, 0, 1, 80963, 1812216, 4798983, 5287506, 3794378, 1863044, 545835, 0, 1, 608832
(list; table; graph; listen)
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OFFSET
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0,6
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COMMENT
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Triangle begins : 1 0, 1 0, 1, 3 0, 1, 12, 13 0, 1, 61, 106, 75 0, 1, 310, 1105, 1035, 541 ... Diagonal is sequence A000670.
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REFERENCES
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A. P. Heinz (1990). Analyse der Grenzen und Moeglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universitaet Freiburg, Freiburg i. Br., Germany.
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LINKS
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Alois P. Heinz (heinz(AT)hs-heilbronn.de), Dec 05 2007, Table of n, a(n) for n = 0..209
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FORMULA
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T(n,0)=A135302(n,0), T(n,k)=A135302(n,k)-A135302(n,k-1) for k>0; e.g.f. of column k=0: tt_0(x)=1, e.g.f. of column k>0: tt_k(x)=t_k(x)-t_{k-1}(x), where t_k(x)=exp(sum_{m=1..k}(x^m/m!*t_{k-m}(x))) if k>=0 and t_k(x)=0 else.
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EXAMPLE
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T(3,3)=13 because there are 13 relations of the given kind for 3 elements:
1R2, 2R1, 1R3, 3R1, 2R3, 3R2;
1R2, 1R3, 2R3, 3R2;
2R1, 2R3, 1R3, 3R1;
3R1, 3R2, 1R2, 2R1;
2R1, 3R1, 2R3, 3R2;
1R2, 3R2, 1R3, 3R1;
1R3, 2R3, 1R2, 2R1;
1R2, 2R3, 1R3;
1R3, 3R2, 1R2;
2R1, 1R3, 2R3;
2R3, 3R1, 2R1;
3R1, 1R2, 3R2;
3R2, 2R1, 3R1;
(the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity)
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MAPLE
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t := proc(k) option remember; if k<0 then 0 else unapply(exp(sum('x^m/m!*t(k-m)(x)', 'm'=1..k)), x) fi; end; tt := proc(k) option remember; unapply((t(k)-t(k-1))(x), x); end; T := proc(n, k) option remember; coeff(series(tt(k)(x), x=0, n+1), x, n)*n!; end; seq(seq(T(n, k), k=0..n), n=0..12);
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CROSSREFS
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Cf. A135302, A000670.
Row sums are in A052880. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 01 2009]
Sequence in context: A112906 A137375 A145881 this_sequence A022695 A067169 A011339
Adjacent sequences: A135310 A135311 A135312 this_sequence A135314 A135315 A135316
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz (heinz(AT)hs-heilbronn.de), Dec 05 2007
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