

A036038


Triangle of multinomial coefficients.


58



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, 210, 420, 630, 840, 1260, 2520, 5040, 1, 8, 28, 56, 70, 56, 168, 280, 420, 560, 336, 840, 1120, 1680, 2520, 1680, 3360, 5040, 6720
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OFFSET

1,3


COMMENTS

The number of terms in the nth row is the number of partitions of n, A000041(n).  Amarnath Murthy, Sep 21 2002
For each n, the partitions are ordered by length and then lexicographically, which is different from the usual practice of ordering all partitions lexicographically.  T. D. Noe, Nov 03 2006
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
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
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) = jth 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


REFERENCES

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".


LINKS

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 8312.
T. Copeland, Witt Differential Generator for Special Jack Symmetric Functions / Polynomials, 2016.
I. Dumitriu, A. Edelman, G. Schuman, MOPS: Multivariate orthogonal polynomials (symbolically), arxiv:0409066 [mathph], 2004.
Wolfdieter Lang, First 10 rows and more.
R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. in Math., 77, p. 76115, 1989.


FORMULA

The nth 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
From Tom Copeland, Nov 26 2016: (Start)
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).
See the Copeland link for more relations between the multinomial coefficients and the Jack symmetric functions. (End)


EXAMPLE

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;


MAPLE

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


MATHEMATICA

Flatten[Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i], Length[ #1]>Length[ #2]&]], {1}], {i, 9}]] (* T. D. Noe, Nov 03 2006 *)


PROG

(Sage)
def A036038_row(n):
return [multinomial(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (1..10): print(A036038_row(n))
# Peter Luschny, Dec 18 2016, updated Nov 02 2019


CROSSREFS

Cf. A036036A036040. Different from A078760. Row sums give A005651.
Cf. A183610 is a table of sums of powers of terms in rows.
Cf. A134264 and A248120.
Cf. A008275, A008276, A094638, A248927.
Cf. A096162 for connections to A130561.
Sequence in context: A010251 A051537 A171999 * A210237 A078760 A103280
Adjacent sequences: A036035 A036036 A036037 * A036039 A036040 A036041


KEYWORD

nonn,easy,nice,tabf,look,hear


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from David W. Wilson and Wouter Meeussen


STATUS

approved



