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A279038
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Triangle of multinomial coefficients read by rows (ordered by decreasing size of the greatest part).
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3
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1, 1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 90, 40, 120, 40, 15, 45, 15, 1, 720, 840, 504, 504, 420, 630, 210, 280, 210, 420, 70, 105, 105, 21, 1, 5040, 5760, 3360, 3360, 2688, 4032, 1344, 1260, 3360, 1260, 2520, 420, 1120, 1120, 1680, 1120, 112, 105, 420, 210, 28, 1
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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The ordering of integer partitions used in this version is also called:
- canonical ordering
- graded reverse lexicographic ordering
- magma (software) ordering
by opposition to the ordering used by Abramowitz and Stegun.
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LINKS
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EXAMPLE
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First rows are:
1
1
1 1
2 3 1
6 8 3 6 1
24 30 20 20 15 10 1
120 144 90 90 40 120 40 15 45 15 1
720 840 504 504 420 630 210 280 210 420 70 105 105 21 1
...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [1],
`if`(i<1, [], [seq(map(x-> x*i^j*j!,
b(n-i*j, i-1))[], j=[iquo(n, i)-t$t=0..n/i])]))
end:
T:= n-> map(x-> n!/x, b(n$2))[]:
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MATHEMATICA
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Flatten[Table[
Map[n!/Times @@ ((First[#]^Length[#]*Factorial[Length[#]]) & /@
Split[#]) &, IntegerPartitions[n]], {n, 1, 8}]]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1},
If[i < 1, {}, Flatten@Table[#*i^j*j!& /@
b[n - i*j, i - 1], {j, Quotient[n, i] - Range[0, n/i]}]]];
T[n_] := n!/#& /@ b[n, n];
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CROSSREFS
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Cf. A000041 (number of partitions of n, length of each row).
Cf. A128628 (triangle of partition lengths)
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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