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%I #112 Apr 30 2022 06:45:47
%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 180,360,720,1,7,21,35,42,105,140,210,210,420,630,840,1260,2520,5040,
%U 1,8,28,56,70,56,168,280,420,560,336,840,1120,1680,2520,1680,3360,5040,6720
%N Triangle of multinomial coefficients.
%C The number of terms in the n-th row is the number of partitions of n, A000041(n). - _Amarnath Murthy_, Sep 21 2002
%C For each n, the partitions are ordered according to A-St: first by length and then lexicographically (arranging the parts in nondecreasing order), which is different from the usual practice of ordering all partitions lexicographically. - _T. D. Noe_, Nov 03 2006
%C For this ordering of the partitions, for n >= 1, see the remarks and the C. F. Hindenburg link given in A036036. - _Wolfdieter Lang_, Jun 15 2012
%C The relation (n+1) * A134264(n+1) = A248120(n+1) / a(n) where the arithmetic is performed for matching partitions in each row n connects the combinatorial interpretations of this array to some topological and algebraic constructs of the two other entries. Also, these seem (cf. MOPS reference, Table 2) to be the coefficients of the Jack polynomial J(x;k,alpha=0). - _Tom Copeland_, Nov 24 2014
%C The conjecture on the Jack polynomials of zero order is true as evident from equation a) on p. 80 of the Stanley reference, suggested to me by Steve Kass. The conventions for denoting the more general Jack polynomials J(n,alpha) vary. Using Stanley's convention, these Jack polynomials are the umbral extensions of the multinomial expansion of (s_1*x_1 + s_2*x_2 + ... + s_(n+1)*x_(n+1))^n in which the subscripts of the (s_k)^j in the symmetric monomial expansions are finally ignored and the exponent dropped to give s_j(alpha) = j-th row polynomial of A094638 or |A008276| in ascending powers of alpha. (The MOPS table has some inconsistency between n = 3 and n = 4.) - _Tom Copeland_, Nov 26 2016
%D Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".
%H David W. Wilson, <a href="/A036038/b036038.txt">Table of n, a(n) for n = 1..11731</a> (rows 1 through 26).
%H Milton Abramowitz and Irene A. Stegun, editors, <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Multinomials: M_1, M_2 and M_3</a>, Handbook of Mathematical Functions, December 1972, pp. 831-2.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2016/11/27/a-note-on-the-jack-symmetric-functions-polynomials/">Witt Differential Generator for Special Jack Symmetric Functions / Polynomials</a>, 2016.
%H I. Dumitriu, A. Edelman, G. Schuman, <a href="http://arxiv.org/abs/math-ph/0409066">MOPS: Multivariate orthogonal polynomials (symbolically)</a>, arxiv:0409066 [math-ph], 2004.
%H Wolfdieter Lang, <a href="/A036038/a036038.pdf">First 10 rows and more.</a>
%H R. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/73.pdf">Some combinatorial properties of Jack symmetric functions</a>, Adv. in Math., 77, p. 76-115, 1989.
%F The n-th row is the expansion of (x_1 + x_2 + ... + x_(n+1))^n in the basis of the monomial symmetric polynomials (m.s.p.). E.g., (x_1 + x_2 + x_3 + x_4)^3 = m[3](x_1,..,x_4) + 3*m[1,2](x_1,..,x_4) + 6*m[1,1,1](x_1,..,x_4) = (Sum_{i=1..4} x_i^3) + 3*(Sum_{i,j=1..4;i != j} x_i^2 x_j) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k). The number of indeterminates can be increased indefinitely, extending each m.s.p., yet the expansion coefficients remain the same. In each m.s.p., unique combinations of exponents and subscripts appear only once with a coefficient of unity. Umbral reduction by replacing x_k^j with x_j in the expansions gives the partition polynomials of A248120. - _Tom Copeland_, Nov 25 2016
%F From _Tom Copeland_, Nov 26 2016: (Start)
%F As an example of the umbral connection to the Jack polynomials: J(3,alpha) = (Sum_{i=1..4} x_i^3)*s_3(alpha) + 3*(Sum_{i,j=1..4;i!=j} x_i^2 x_j)*s_2(alpha)*s_1(alpha)+ 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k)*s_1(alpha)*s_1(alpha)*s_1(alpha) = (Sum_{i=1..4} x_i^3)*(1+alpha)*(1+2*alpha)+ 3*(sum_{i,j=1..4;i!=j} x_i^2 x_j)*(1+alpha) + 6*(Sum_{i,j,k=1..4;i < j < k} x_i x_j x_k).
%F See the Copeland link for more relations between the multinomial coefficients and the Jack symmetric functions. (End)
%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;
%p nmax:=7: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while s<n do j:=j+1: s:=s+B(m)[j]: x(j):=B(m)[j]: end do; jmax:=j; for r from 1 to n do q(r):=0 od: for r from 1 to n do for j from 1 to jmax do if x(j)=r then q(r):=q(r)+1 fi: od: od: A036038(n, m) := n!/ (mul((t!)^q(t), t=1..n)); od: od: seq(seq(A036038(n, m), m=1..numbpart(n)), n=1..nmax); # _Johannes W. Meijer_, Jul 14 2016
%t Flatten[Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i], Length[ #1]>Length[ #2]&]], {1}], {i,9}]] (* _T. D. Noe_, Nov 03 2006 *)
%o (Sage)
%o def ASPartitions(n, k):
%o Q = [p.to_list() for p in Partitions(n, length=k)]
%o for q in Q: q.reverse()
%o return sorted(Q)
%o def A036038_row(n):
%o return [multinomial(p) for k in (0..n) for p in ASPartitions(n, k)]
%o for n in (1..10): print(A036038_row(n))
%o # _Peter Luschny_, Dec 18 2016, corrected Apr 30 2022
%Y Cf. A036036-A036040. Different from A078760. Row sums give A005651.
%Y Cf. A183610 is a table of sums of powers of terms in rows.
%Y Cf. A134264 and A248120.
%Y Cf. A008275, A008276, A094638, A248927.
%Y Cf. A096162 for connections to A130561.
%K nonn,easy,nice,tabf,look,hear
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_ and _Wouter Meeussen_
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