A353132
Triangle read by rows of partial Bell polynomials B_{n,k}(x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.
Original entry on oeis.org
2, 1, 4, 2, 6, 8, 3, 18, 24, 16, 4, 40, 100, 80, 32, 5, 78, 305, 440, 240, 64, 6, 140, 798, 1750, 1680, 672, 128, 7, 236, 1876, 5838, 8400, 5824, 1792, 256, 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512, 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024
Offset: 1
For n = 4, k = 2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3) = 4*x_1*x_3 + 3*x_2^2, so T(4,2) = B_{4,2}(2,2,12) - (4*2*12 + 3*2^2)/3! = 18.
Triangle begins:
[1] 2;
[2] 1, 4;
[3] 2, 6, 8;
[4] 3, 18, 24, 16;
[5] 4, 40, 100, 80, 32;
[6] 5, 78, 305, 440, 240, 64;
[7] 6, 140, 798, 1750, 1680, 672, 128;
[8] 7, 236, 1876, 5838, 8400, 5824, 1792, 256;
[9] 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512;
[10] 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024.
- Jordan Weaver, Rows 1 to 40 of triangle, flattened
- E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.
- E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
- Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
- A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
- A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.
A335256
Irregular triangle read by rows: row n gives the coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ..., x_n) with monomials sorted into standard order.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 4, 3, 1, 1, 10, 10, 15, 5, 10, 1, 1, 15, 20, 45, 15, 60, 15, 6, 15, 10, 1, 1, 21, 35, 105, 35, 210, 105, 21, 105, 70, 105, 7, 21, 35, 1, 1, 28, 56, 210, 70, 560, 420, 56, 420, 280, 840, 105, 28, 168, 280, 210, 280, 8, 28, 56, 35, 1
Offset: 1
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] + 15*x[2]^3 + 6*x[1]*x[5] + 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] + 105*x[1]*x[2]^3 + 21*x[1]^2*x[5] + 105*x[1]*x[2]*x[4] + 70*x[1]*x[3]^2 + 105*x[2]^2*x[3] + 7*x[1]*x[6] + 21*x[2]*x[5] + 35*x[3]*x[4] + 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, 15, 6, 15, 10, 1;
1, 21, 35, 105, 35, 210, 105, 21, 105, 70, 105, 7, 21, 35, 1;
...
- L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 134 and 307-310.
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, Chapter 2, Section 8 and table on page 49.
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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(8); # Peter Luschny, May 30 2020
-
imax = 10;
polys = (CoefficientList[Exp[Sum[x[i]*t^i/i!, {i, 1, imax}]] + O[t]^imax // Normal, t]*Range[0, imax-1]!) // Rest;
Table[MonomialList[polys[[i]], Array[x, i], "DegreeLexicographic"] /. x[] -> 1, {i, 1, imax-1}] // Flatten (* _Jean-François Alcover, Jun 02 2024 *)
-
/* It produces the partial exponential Bell polynomials in decreasing degree, but the monomials are not necessarily in standard order. */
Bell(n,k)= { my(x, v, dv, var = i->eval(Str("X", i))); v = vector(n, i, if (i==1, 'E, var(i-1))); dv = vector(n, i, if (i==1, 'X*var(1)*'E, var(i))); x = diffop('E, v, dv, n) / 'E; if (k < 0, subst(x,'X, 1), polcoeff(x, k, 'X)); };
row(n) = for(k=1, n, print1("[", Bell(n, n+1-k), "]", ","))
A356656
Partition triangle read by rows. The coefficients of the incomplete Bell polynomials.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 4, 3, 6, 1, 0, 1, 5, 10, 10, 15, 10, 1, 0, 1, 6, 15, 10, 15, 60, 15, 20, 45, 15, 1, 0, 1, 7, 21, 35, 21, 105, 70, 105, 35, 210, 105, 35, 105, 21, 1, 0, 1, 8, 28, 56, 35, 28, 168, 280, 210, 280, 56, 420, 280, 840, 105, 70, 560, 420, 56, 210, 28, 1
Offset: 0
The triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 1, 3, 1;
[4] 0, 1, [4, 3], 6, 1;
[5] 0, 1, [5, 10], [10, 15], 10, 1;
[6] 0, 1, [6, 15, 10], [15, 60, 15], [20, 45], 15, 1;
[7] 0, 1, [7, 21, 35], [21, 105, 70, 105], [35, 210, 105], [35, 105], 21, 1;
Summing the bracketed terms reduces the triangle to A048993.
The first few polynomials are:
[0] 1;
[1] 0, z[0];
[2] 0, z[1], z[0]^2;
[3] 0, z[2], 3*z[0]*z[1], z[0]^3;
[4] 0, z[3], 4*z[0]*z[2]+3*z[1]^2, 6*z[0]^2*z[1], z[0]^4;
[5] 0, z[4], 5*z[0]*z[3]+10*z[1]*z[2], 10*z[0]^2*z[2]+15*z[0]*z[1]^2, 10*z[0]^3* z[1], z[0]^5;
It is noteworthy that the substitution z[n] -> n! for n >= 0 yields A132393. More examples are given in the authors blog post (see links).
-
aRow := n -> seq(coeffs(IncompleteBellB(n, k, seq(z[i], i = 0..n))), k = 0..n):
seq(aRow(n), n = 0..8);
-
from functools import cache
@cache
def incomplete_bell_polynomial(n, k):
Z = var(["z_" + str(i) for i in range(n - k + 1)])
R = PolynomialRing(ZZ, Z, n - k + 1, order='lex')
if k == 0: return R(k^n)
return R(sum(binomial(n-1,j-1) * incomplete_bell_polynomial(n-j,k-1) * Z[j-1]
for j in range(n - k + 2)).expand())
def poly_row(n): return [incomplete_bell_polynomial(n, k) for k in range(n + 1)]
def coeff_row(n): return flatten([[0] if (c := p.coefficients()) == [] else c for p in poly_row(n)])
for n in range(8): print(coeff_row(n))
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