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A210237 Triangle of distinct values M(n) of multinomial coefficients for partitions of n in increasing order of n and M(n). 3
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, 180, 360, 720, 1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, 168, 280, 336, 420, 560, 840, 1120, 1680, 2520, 3360, 5040, 6720 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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.
There is no known formula for M(n) sequence, however the asymptotic behavior has been studied, see the paper by Andrews, Knopfmacher, and Zimmermann.
The number of terms per row (for each value of n starting with n=1) forms sequence A070289.
LINKS
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, arXiv:math/0509470 [math.CO], 2005.
Sergei Viznyuk, C-program for the sequence
EXAMPLE
Trianglebegins:
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, 180, 360, 720;
1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040;
...
Thus for n=4 (fourth row) the distinct values of multinomial coefficients are:
4!/(4!) = 1
4!/(3!1!) = 4
4!/(2!2!) = 6
4!/(2!1!1!) = 12
4!/(1!1!1!1!) = 24
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i<2, {1},
{seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=0..n/i)})
end:
T:= n-> sort([map(x-> n!/x, b(n, n))[]])[]:
seq(T(n), n=1..10); # Alois P. Heinz, Aug 13 2012
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A338797 A171999 A036038 * A078760 A348113 A103280
KEYWORD
nonn,tabf
AUTHOR
Sergei Viznyuk, Mar 18 2012
STATUS
approved

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Last modified July 19 02:27 EDT 2024. Contains 374388 sequences. (Running on oeis4.)