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A279038 Triangle of multinomial coefficients read by rows (ordered by decreasing size of the greatest part).

%I A279038 #20 Jun 01 2021 08:05:26
%S A279038 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,
%T A279038 15,45,15,1,720,840,504,504,420,630,210,280,210,420,70,105,105,21,1,
%U A279038 5040,5760,3360,3360,2688,4032,1344,1260,3360,1260,2520,420,1120,1120,1680,1120,112,105,420,210,28,1
%N A279038 Triangle of multinomial coefficients read by rows (ordered by decreasing size of the greatest part).
%C A279038 The ordering of integer partitions used in this version is also called:
%C A279038 - canonical ordering
%C A279038 - graded reverse lexicographic ordering
%C A279038 - magma (software) ordering
%C A279038 by opposition to the ordering used by Abramowitz and Stegun.
%H A279038 Alois P. Heinz, <a href="/A279038/b279038.txt">Rows n = 0..28, flattened</a>
%e A279038 First rows are:
%e A279038     1
%e A279038     1
%e A279038     1   1
%e A279038     2   3   1
%e A279038     6   8   3   6   1
%e A279038    24  30  20  20  15   10   1
%e A279038   120 144  90  90  40  120  40  15  45  15  1
%e A279038   720 840 504 504 420  630 210 280 210 420 70 105 105 21 1
%e A279038   ...
%p A279038 b:= proc(n, i) option remember; `if`(n=0, [1],
%p A279038       `if`(i<1, [], [seq(map(x-> x*i^j*j!,
%p A279038        b(n-i*j, i-1))[], j=[iquo(n, i)-t$t=0..n/i])]))
%p A279038     end:
%p A279038 T:= n-> map(x-> n!/x, b(n$2))[]:
%p A279038 seq(T(n), n=0..10);  # _Alois P. Heinz_, Dec 04 2016
%t A279038 Flatten[Table[
%t A279038   Map[n!/Times @@ ((First[#]^Length[#]*Factorial[Length[#]]) & /@
%t A279038         Split[#]) &, IntegerPartitions[n]], {n, 1, 8}]]
%t A279038 (* Second program: *)
%t A279038 b[n_, i_] := b[n, i] = If[n == 0, {1},
%t A279038      If[i < 1, {}, Flatten@Table[#*i^j*j!& /@
%t A279038      b[n - i*j, i - 1], {j, Quotient[n, i] - Range[0, n/i]}]]];
%t A279038 T[n_] := n!/#& /@ b[n, n];
%t A279038 T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, Jun 01 2021, after _Alois P. Heinz_ *)
%Y A279038 Cf. A000041 (number of partitions of n, length of each row).
%Y A279038 Cf. A128628 (triangle of partition lengths)
%Y A279038 Cf. A036039 (a different ordering), A102189 (row reversed version of A036039).
%Y A279038 Row sums give A000142.
%K A279038 nonn,tabf,look,easy
%O A279038 0,5
%A A279038 _David W. Wilson_ and _Olivier Gérard_, Dec 04 2016
%E A279038 One term for row n=0 prepended by _Alois P. Heinz_, Dec 04 2016