A279038 - OEIS

%I #20 Jun 01 2021 08:05:26

%S 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 15,45,15,1,720,840,504,504,420,630,210,280,210,420,70,105,105,21,1,

%U 5040,5760,3360,3360,2688,4032,1344,1260,3360,1260,2520,420,1120,1120,1680,1120,112,105,420,210,28,1

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

%C The ordering of integer partitions used in this version is also called:

%C - canonical ordering

%C - graded reverse lexicographic ordering

%C - magma (software) ordering

%C by opposition to the ordering used by Abramowitz and Stegun.

%H Alois P. Heinz, <a href="/A279038/b279038.txt">Rows n = 0..28, flattened</a>

%e First rows are:

%e     1

%e     1

%e     1   1

%e     2   3   1

%e     6   8   3   6   1

%e    24  30  20  20  15   10   1

%e   120 144  90  90  40  120  40  15  45  15  1

%e   720 840 504 504 420  630 210 280 210 420 70 105 105 21 1

%e   ...

%p b:= proc(n, i) option remember; `if`(n=0, [1],

%p       `if`(i<1, [], [seq(map(x-> x*i^j*j!,

%p        b(n-i*j, i-1))[], j=[iquo(n, i)-t$t=0..n/i])]))

%p     end:

%p T:= n-> map(x-> n!/x, b(n$2))[]:

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Dec 04 2016

%t Flatten[Table[

%t   Map[n!/Times @@ ((First[#]^Length[#]*Factorial[Length[#]]) & /@

%t         Split[#]) &, IntegerPartitions[n]], {n, 1, 8}]]

%t (* Second program: *)

%t b[n_, i_] := b[n, i] = If[n == 0, {1},

%t      If[i < 1, {}, Flatten@Table[#*i^j*j!& /@

%t      b[n - i*j, i - 1], {j, Quotient[n, i] - Range[0, n/i]}]]];

%t T[n_] := n!/#& /@ b[n, n];

%t T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, Jun 01 2021, after _Alois P. Heinz_ *)

%Y Cf. A000041 (number of partitions of n, length of each row).

%Y Cf. A128628 (triangle of partition lengths)

%Y Cf. A036039 (a different ordering), A102189 (row reversed version of A036039).

%Y Row sums give A000142.

%K nonn,tabf,look,easy

%O 0,5

%A _David W. Wilson_ and _Olivier Gérard_, Dec 04 2016

%E One term for row n=0 prepended by _Alois P. Heinz_, Dec 04 2016