|
%I #33 Aug 26 2019 17:53:31
%S 1,1,2,1,3,6,1,4,6,12,24,1,5,10,20,30,60,120,1,6,15,20,30,60,90,120,
%T 180,360,720,1,7,21,35,42,105,140,210,420,630,840,1260,2520,5040,1,8,
%U 28,56,70,168,280,336,420,560,840,1120,1680,2520,3360,5040,6720
%N Triangle of distinct values M(n) of multinomial coefficients for partitions of n in increasing order of n and M(n).
%C Differs from A036038 after a(37). To illustrate where the difference comes from, consider 4,1,1,1 and 3,2,2 are two different partitions of 7 having the same value of multinomial coefficient M(n)=n!/(m1!*m2!*...*mk!)=210.
%C There is no known formula for M(n) sequence, however the asymptotic behavior has been studied, see the paper by Andrews, Knopfmacher, and Zimmermann.
%C The number of terms per row (for each value of n starting with n=1) forms sequence A070289.
%H Alois P. Heinz, <a href="/A210237/b210237.txt">Rows n = 1..29, flattened</a>
%H George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, <a href="http://arxiv.org/abs/math/0509470">On the Number of Distinct Multinomial Coefficients</a>, arXiv:math/0509470 [math.CO], 2005.
%H Sergei Viznyuk, <a href="http://phystech.com/ftp/s_A210237.c">C-program</a> for the sequence
%e Trianglebegins:
%e 1;
%e 1, 2;
%e 1, 3, 6;
%e 1, 4, 6, 12, 24;
%e 1, 5, 10, 20, 30, 60, 120;
%e 1, 6, 15, 20, 30, 60, 90, 120, 180, 360, 720;
%e 1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040;
%e ...
%e Thus for n=4 (fourth row) the distinct values of multinomial coefficients are:
%e 4!/(4!) = 1
%e 4!/(3!1!) = 4
%e 4!/(2!2!) = 6
%e 4!/(2!1!1!) = 12
%e 4!/(1!1!1!1!) = 24
%p b:= proc(n, i) option remember; `if`(n=0 or i<2, {1},
%p {seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=0..n/i)})
%p end:
%p T:= n-> sort([map(x-> n!/x, b(n, n))[]])[]:
%p seq(T(n), n=1..10); # _Alois P. Heinz_, Aug 13 2012
%t b[n_, i_] := b[n, i] = If[n==0 || i<2, {1}, Union[Flatten @ Table[(#*i!^j&) /@ b[n-i*j, i-1], {j, 0, n/i}]]]; T[n_] := Sort[Flatten[n!/#& /@ b[n, n]] ]; Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Feb 05 2017, after _Alois P. Heinz_ *)
%Y Cf. A036038, A210238, A078760, A209936, A080577, A070289.
%K nonn,tabf
%O 1,3
%A _Sergei Viznyuk_, Mar 18 2012
|
