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A178867 Irregular triangle read by rows: multinomial coefficients, version 3; alternatively, row n gives coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into some unknown order. 14
1, 1, 1, 1, 3, 1, 1, 6, 4, 3, 1, 1, 10, 10, 15, 5, 10, 1, 1, 15, 20, 45, 15, 60, 6, 15, 15, 10, 1, 1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 35, 21, 1, 1, 28, 56, 210, 70, 560, 56, 420, 420, 280, 28, 840, 280, 168, 8, 280, 210, 105, 56, 35, 28, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
"Exponential" here means in contrast to "ordinary", cf. A263633 (see Comtet). "Standard order" means as produced by Maple's "sort" command. - N. J. A. Sloane, Oct 28 2015
From Petros Hadjicostas, May 27 2020: (Start)
According to the Maple help files for the "sort" command, polynomials in multiple variables are "sorted in total degree with ties broken by lexicographic order (this is called graded lexicographic order)."
Thus, for example, x_1^2*x_3 = x_1*x_1*x_3 > x_1*x_2*x_2 = x_1*x_2^2, while x_1^2*x_4 = x_1*x_1*x_4 > x_1*x_2*x_3.
It appears that the authors' n-th row does give the coefficients of the n-th complete exponential Bell polynomial. Starting with row 6, however, it is unknown in what order the monomials of the n-th complete exponential Bell polynomial follow. It is definitely not the standard order nor any other known order. (End)
This version of the multinomial coefficients was discovered while calculating the probability that two 23 year old chessplayers would play each other on their birthday during a Dutch Chess Championship. This unique encounter took place on Apr 05 2008. Its probability is 1 in 50000 years, see the Meijer-Nepveu article.
The third version of the multinomial coefficients can be constructed with the basic multinomial coefficients A178866; see the formulas below. These multinomial coefficients appear in the columns of the multinomial coefficient matrix MCM[n,m] (n >= 1 and m >= 1).
We observe that the sum of the C(m,n) coefficients follow the A000296(n) sequence. The missing C(m, n=1) corresponds to A000296(1) = 0.
The number of multinomial coefficients in a triangle row leads to the partition numbers A000041. The row sums lead to the Bell numbers A000110.
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 134, 307-310.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, Chapter 2, Section 8 and table on page 49.
LINKS
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021. Mentions this sequence p. 27.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, 4(1) (2008), 176-187.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
FORMULA
G.f.: Exp(Sum_{i >= 1} x_i*t^i/i!) = 1 + Sum_{n >= 1} B_n(x_1, x_2,...)*t^n/n!. [Comtet, p. 134, Eq. [3b]. - N. J. A. Sloane, Oct 28 2015]
For m >= 1, the formulas for the first few matrix columns are:
MCM[1,m] = A178866(1)*C(m,0) = 1*C(m,0);
MCM[2,m] = A178866(2)*C(m,2) = 1*C(m,2);
MCM[3,m] = A178866(3)*C(m,3) = 1*C(m,3);
MCM[4,m] = A178866(4)*C(m,4) = 3*C(m,4) and
MCM[5,m] = A178866(5)*C(m,4) = 1*C(m,4);
MCM[6,m] = A178866(6)*C(m,5) = 10*C(m,5) and
MCM[7,m] = A178866(7)*C(m,5) = 1*C(m,5);
MCM[8,m] = A178866(8)*C(m,6) = 15*C(m,6) and
MCM[9,m] = A178866(9)*C(m,6) = 15*C(m,6) and
MCM[10,m] = A178866(10)*C(m,6) = 10*C(m,6) and
MCM[11,m] = A178866(11)*C(m,6) = 1*C(m,6).
EXAMPLE
The first few complete exponential Bell polynomials are:
(1) x[1];
(2) x[1]^2 + x[2];
(3) x[1]^3 + 3*x[1]*x[2] + x[3];
(4) x[1]^4 + 6*x[1]^2*x[2] + 4*x[1]*x[3] + 3*x[2]^2 + x[4];
(5) x[1]^5 + 10*x[1]^3*x[2] + 10*x[1]^2*x[3] + 15*x[1]*x[2]^2 + 5*x[1]*x[4] + 10*x[2]*x[3] + x[5];
(6) x[1]^6 + 15*x[1]^4*x[2] + 20*x[1]^3*x[3] + 45*x[1]^2*x[2]^2 + 15*x[1]^2*x[4] + 60*x[1]*x[2]*x[3] + 6*x[1]*x[5] + 15*x[2]^3 + 15*x[2]*x[4] + 10*x[3]^2 + x[6].
(7) x[1]^7 + 21*x[1]^5*x[2] + 35*x[1]^4*x[3] + 105*x[1]^3*x[2]^2 + 35*x[1]^3*x[4] + 210*x[1]^2*x[2]*x[3] + 21*x[1]^2*x[5] + 105*x[1]*x[2]^3 + 105*x[1]*x[2]*x[4] + 70*x[1]*x[3]^2 + 7*x[1]*x[6] + 105*x[2]^2*x[3] + 35*x[3]*x[4] + 21*x[2]*x[5] + x[7].
...
The first few rows of the triangle are
1;
1, 1;
1, 3, 1;
1, 6, 4, 3, 1;
1, 10, 10, 15, 5, 10, 1;
1, 15, 20, 45, 15, 60, 6, 15, 15, 10, 1;
1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 35, 21, 1;
...
MAPLE
with(combinat): nmax:=11; A178866(1):=1: T:=1: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): k:=0: for r from 1 to y(n) do if P(n)[r, 1]>1 then k:=k+1; B(k):=P(n)[r]: fi; od: A002865(n):=k; for k from 1 to A002865(n) do s:=0: j:=1: while s<n do s:=s+B(k)[j]: x(j+1):=B(k)[j]: j:=j+1; end do; jmax:=j; for r from 1 to n do q(r):=0 od: for r from 2 to n do for j from 1 to jmax do if x(j)=r then q(r):=q(r)+1 fi: od: od: M3[n, k]:= n!/(product((t!)^q(t)*q(t)!, t=1..n)): od: a:=sort([seq(M3[n, k], k=1..A002865(n))], `>`): for k from 1 to A002865(n) do M3[n, k]:=a[k] od: for k from 1 to A002865(n) do T:=T+1: A178866(T):= M3[n, k]: od: od:
mmax:=nmax: n:=1: for m from 1 to mmax do MCM[n, m]:= A178866(n) od: n:=2: r:=1: for i from 2 to nmax do p:=A002865(i): r:=r+1: while p>0 do for m from 1 to mmax do MCM[n, m]:=A178866(n)*binomial(m, r) od: p:=p-1: n:=n+1: od: od: T:=0: for m from 1 to mmax do for n from 1 to numbpart(m) do T:=T+1; A178867(T):= MCM[n, m]; od: od; seq(A178867(n), n=1..T);
# To produce the complete exponential Bell polynomials in standard order, from N. J. A. Sloane, Oct 28 2015
M:=12;
EE:=add(x[i]*t^i/i!, i=1..M);
t1:=exp(EE);
t2:=series(t1, t, M);
Q:=k->sort(expand(k!*coeff(t2, t, k)));
for k from 1 to 8 do lprint(k, Q(k)); od;
# To produce the coefficients of the complete exponential Bell polynomials in standard order:
triangle := proc(numrows) local E, s, Q;
E := add(x[i]*t^i/i!, i=1..numrows);
s := series(exp(E), t, numrows+1);
Q := k -> sort(expand(k!*coeff(s, t, k)));
seq(print(coeffs(Q(k))), k=1..numrows) end:
triangle(6); # Peter Luschny, May 27 2020
CROSSREFS
Cf. A036040 (version 1 of multinomial coefficients), A080575 (version 2).
Sequence in context: A211351 A124802 A211350 * A335256 A102036 A121524
KEYWORD
easy,nonn,tabf
AUTHOR
Johannes W. Meijer and Manuel Nepveu (Manuel.Nepveu(AT)tno.nl), Jun 21 2010, Jun 24 2010
EXTENSIONS
Alternative definition as coefficients of complete Bell polynomials added by N. J. A. Sloane, Oct 28 2015
Various sections and name edited by Petros Hadjicostas, May 28 2020
STATUS
approved

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Last modified May 19 14:23 EDT 2024. Contains 372697 sequences. (Running on oeis4.)