A263633 Irregular triangle read by rows: row n gives coefficients of n-th ordinary Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into graded lexicographic order.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 3, 3, 2, 2, 1, 1, 5, 4, 6, 3, 6, 1, 2, 2, 1, 1, 1, 6, 5, 10, 4, 12, 4, 3, 6, 3, 3, 2, 2, 2, 1, 1, 7, 6, 15, 5, 20, 10, 4, 12, 6, 12, 1, 3, 6, 6, 3, 3, 2, 2, 2, 1, 1, 1, 8, 7, 21, 6, 30, 20, 5, 20, 10, 30, 5, 4, 12, 12, 12, 12, 4, 3, 6, 6, 3, 3, 6, 1, 2, 2, 2, 2, 1

Offset: 1

Author

N. J. A. Sloane, Oct 28 2015

Keywords

• nonn
• tabf

Comments

"Ordinary" here means in contrast to "exponential", cf. A178867 (see Comtet).

Graded lexicographic order with x[1] > x[2] > ... > x[n] means that monomials are compared first by their total degree, with ties broken by lexicographic order. These monomials correspond to integer partitions.

Row sums are powers of 2. Numbers of terms in rows are partition numbers A000041.

OP_n(-a_1,..,-a_n) = EP_n(a_1,2!*a_2,..,n!*a_n) / n!, where OP_n(a_1,..,a_n) are the partition polynomials of this entry and EP_n, the polynomials of A133314; i.e., the sequences are related as reciprocal o.g.f.s are to reciprocal e.g.f.s. The polynomials play a role in expansion of the iterated Lie derivative (g(x) D_x)^n) formalism for the compositional inversion sketched in A133932. With x[n] = t, the array reduces to the Pascal matrix A007318. - Tom Copeland, Sep 19 2016

The signed row partition polynomials can be generated by the Gram determinants of equation 2.23 on page 133 of the Verde-Star paper. E.g., h_3 = -b_1^3 + 2 b_1 b_2 - b_3 corresponds to the third row. The connection to A133314 is obtained by substituting a(k) = k!*b_k = -k!*x[k] and b(k) = k!*h_k in A133314 to compute reciprocals of o.g.f.s rather than e.g.f.s. - Tom Copeland, Dec 04 2016

For a relation to lambda operations in K-theory on vector bundles, see p. 218 of Dugger. - Tom Copeland, Jul 25 2017

Since E(x) = (1+x_1*x)(1+x_2*x)...(1+x_m*x) is the o.g.f. for the elementary symmetric polynomials e_n(x_1,x_2,...,x_m) and the o.g.f. for the complete symmetric polynomials h_n(x_1,x_2,...,x_m) is H(x) = 1 / E(-x), this entry's partition polynomials with correct signs give either sequence in terms of the other. - Tom Copeland, Jan 29 2018

A133314 has an interpretation as weighted surjective mappings. With the connections of this mapping colored and permuted to give mappings distinguished by the order of the colorings (an induced linear ordering by color of the connecting arrows), the signed partition polynomials of this entry, multiplied by n!, are generated. - Tom Copeland, Sep 10 2020

Examples

			The first few polynomials are:
1, x[1]
2, x[1]^2 + x[2]
3, x[1]^3 + 2*x[1]*x[2] + x[3]
4, x[1]^4 + 3*x[1]^2*x[2] + 2*x[1]*x[3] + x[2]^2 + x[4]
5, x[1]^5 + 4*x[1]^3*x[2] + 3*x[1]^2*x[3] + 3*x[1]*x[2]^2 + 2*x[1]*x[4] + 2*x[2]*x[3] + x[5]
6, x[1]^6 + 5*x[1]^4*x[2] + 4*x[1]^3*x[3] + 6*x[1]^2*x[2]^2 + 3*x[1]^2*x[4] + 6*x[1]*x[2]*x[3] + x[2]^3 + 2*x[1]*x[5] + 2*x[2]*x[4] + x[3]^2 + x[6]
...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 136, 309.

Links

• Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015; In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019; Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
• D. Dugger, A Geometric Introduction to K-Theory.
• Luis Verde-Star, Representation of symmetric functions as Gram determinants, Advances in Mathematics, 1 Dec 1998, Vol. 140(1):128-143.
• Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

Crossrefs

For triangle of coefficients of exponential Bell polynomials see A178867.

Cf. A000041, A263634.

Cf. A007318, A133314, A133932.

Programs

Maple

with(Groebner):
A263633_row := proc(n) local EE,t1,t2,Q,F,X,p,L,q,c,r;
EE := add(x[i]*t^i, i=1..2*n);
t1 := 1/(1-EE):
t2 := series(t1, t, 2*n):
Q := k -> expand(coeff(t2, t, k));
X := seq(x[i], i=1..n);
p := Q(n);
L := [];
while p <> 0 do
   r := LeadingTerm(p, grlex(X));
   c := r[1]; q := r[2];
   p := p - c*q;
   L := [op(L), c];
od;
L end:
for n from 1 to 8 do A263633_row(n) od; # Program expanded by Peter Luschny, Sep 26 2016

Formula

G.f.: 1/(1-Sum_{i >= 1} x_i*t^i) = 1 + Sum_{n >= 1} B_n(x_1, x_2,...)*t^n. [Comtet, p. 136, Eq. [3o'].]

Extensions

More terms and some edits by Peter Luschny, Sep 26 2016