A268441 -id:A268441 - cp's OEIS frontend

A269941 Triangle read by rows, the coefficients of the partial P-polynomials.

1, 0, -1, 0, -1, 1, 0, -1, 2, -1, 0, -1, 1, 2, -3, 1, 0, -1, 2, 2, -3, -3, 4, -1, 0, -1, 1, 2, 2, -1, -6, -3, 6, 4, -5, 1, 0, -1, 2, 2, 2, -3, -3, -6, -3, 4, 12, 4, -10, -5, 6, -1, 0, -1, 1, 2, 2, 2, -3, -3, -6, -6, -3, 1, 12, 6, 12, 4, -10, -20, -5, 15, 6, -7, 1

Offset: 0

Peter Luschny, Mar 08 2016

For the definition of the partial P-polynomials see the link 'P-transform'. The triangle of coefficients of the inverse partial P-polynomials is A269942.

			[[1]],
[[0], [-1]],
[[0], [-1], [1]],
[[0], [-1], [2], [-1]],
[[0], [-1], [1, 2], [-3], [1]],
[[0], [-1], [2, 2], [-3, -3], [4], [-1]],
[[0], [-1], [1, 2, 2], [-1, -6, -3], [6, 4], [-5], [1]],
[[0], [-1], [2, 2, 2], [-3, -3, -6, -3], [4, 12, 4], [-10, -5], [6], [-1]]
Replacing the sublists by their sums reduces the triangle to a signed version of the triangle A097805.

Peter Luschny, The P-transform, 2016.
Peter Luschny, The Partition Transform, A SageMath Jupyter Notebook, GitHub, 2016/2022.
Marko Riedel, Answer to Question 4943578, Mathematics Stack Exchange, 2024.
Peter Taylor, Answer to Question 474483, MathOverflow, 2024.

Cf. A097805, A268441, A268442, A269942.

PTrans := proc(n, f, nrm:=NULL) local q, p, r, R;
if n = 0 then return [1] fi; R := [seq(0,j=0..n)];
for q in combinat:-partition(n) do
   p := [op(ListTools:-Reverse(q)),0]; r := p[1]+1;
   mul(binomial(p[j], p[j+1])*f(j)^p[j], j=1..nops(q));
   R[r] := R[r]-(-1)^r*% od;
if nrm = NULL then R else [seq(nrm(n,k)*R[k+1],k=0..n)] fi end:
A269941_row := n -> seq(coeffs(p), p in PTrans(n, n -> x[n])):
seq(lprint(A269941_row(n)), n=0..8);

def PtransMatrix(dim, f, norm = None, inverse = False, reduced = False):
    i = 1; F = [1]
    if reduced:
        while i <= dim: F.append(f(i)); i += 1
    else:
        while i <= dim: F.append(F[i-1]*f(i)); i += 1
    C = [[0 for k in range(m+1)] for m in range(dim)]
    C[0][0] = 1
    if inverse:
        for m in (1..dim-1):
            C[m][m] = -C[m-1][m-1]/F[1]
            for k in range(m-1, 0, -1):
                C[m][k] = -(C[m-1][k-1]+sum(F[i]*C[m][k+i-1]
                          for i in (2..m-k+1)))/F[1]
    else:
        for m in (1..dim-1):
            C[m][m] = -C[m-1][m-1]*F[1]
            for k in range(m-1, 0, -1):
                C[m][k] = -sum(F[i]*C[m-i][k-1] for i in (1..m-k+1))
    if norm == None: return C
    for m in (1..dim-1):
        for k in (1..m): C[m][k] *= norm(m,k)
    return C
def PMultiCoefficients(dim, norm = None, inverse = False):
    def coefficient(p):
        if p <= 1: return [p]
        return SR(p).fraction(ZZ).numerator().coefficients()
    f = lambda n: var('x'+str(n))
    P = PtransMatrix(dim, f, norm, inverse)
    return [[coefficient(p) for p in L] for L in P]
print(flatten(PMultiCoefficients(9)))

A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 3, -1, -3, 1, 0, -15, 10, -1, 15, -4, -6, 1, 0, 105, -105, 10, 15, -1, -105, 60, -5, 45, -10, -10, 1, 0, -945, 1260, -280, -210, 35, 21, -1, 945, -840, 70, 105, -6, -420, 210, -15, 105, -20, -15, 1

Offset: 0

Peter Luschny, Feb 06 2016

The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the link 'Bell transform'.

			[[1]],
[[0], [1]],
[[0], [-1],                    [1]],
[[0], [3, -1],                 [-3],           [1]],
[[0], [-15, 10, -1],           [15, -4],       [-6],      [1]],
[[0], [105, -105, 10, 15, -1], [-105, 60, -5], [45, -10], [-10], [1]]
Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of first kind (A048994). The column 1 of sublists is A176740 (missing the leading 1) and A134685 in different order.

Peter Luschny, First 21 rows, flattened
Peter Luschny, The Bell transform
Peter Luschny, SageMath implementation

Cf. A036040, A048994, A134685, A176740, A268441.

A268442Matrix[dim_] := Module[ {v, r, A},
v = Table[Subscript[x,j],{j,1,dim}];
r = Table[Subscript[x,j]->1,{j,1,n}];
A = Table[Table[BellY[n,k,v], {k,0,dim}], {n,0,dim}];
Table[Table[MonomialList[Inverse[A][[n,k]]/. r[[1]],
v, Lexicographic] /. r, {k,1,n}] // Flatten, {n,1,dim}]];
A268442Matrix[7] // Flatten

Sage
```
# see link
```

A353131 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)!, n>=1, 1<=k<=n.

Original entry on oeis.org

2, 2, 4, 12, 12, 8, 72, 108, 48, 16, 480, 960, 600, 160, 32, 3600, 9360, 7320, 2640, 480, 64, 30240, 100800, 95760, 42000, 10080, 1344, 128, 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256, 2903040, 15240960, 20442240, 12337920, 4142880, 854784, 112896, 9216, 512

Offset: 1

Jordan Weaver, Apr 24 2022

			For n=4,k=2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3)=4x_1x_3+3x_2^2, so a(4,2)=B_{4,2}(2,2,12)=4*2*12+3*2^2=108.
Triangle starts:
[1]      2;
[2]      2,       4;
[3]     12,      12,       8;
[4]     72,     108,      48,     16;
[5]    480,     960,     600,    160,     32;
[6]   3600,    9360,    7320,   2640,    480,    64;
[7]  30240,  100800,   95760,  42000,  10080,  1344,  128;
[8] 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256.

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.

Cf. A000079, A353132, A349413, A268441, A178867.

T(n,k) = A353132(n,k)*(n-k+1)!.

Sum_{k=1..n} T(n,k)/(n-k+1)! = A349458(n).

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

Jordan Weaver, Apr 24 2022

			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.

Cf. A000079, A353131, A349413, A268441, A178867.

T(n,k) = A353131(n,k)/(n-k+1)!

Sum_{k=1..n} T(n,k) = A349458(n).

A269942 Triangle read by rows, the coefficients of the inverse partial P-polynomials.

Original entry on oeis.org

1, 0, -1, 0, -1, 1, 0, -2, 1, 2, -1, 0, -5, 5, -1, 5, -2, -3, 1, 0, -14, 21, -3, -6, 1, 14, -12, 2, -9, 3, 4, -1, 0, -42, 84, -28, -28, 7, 7, -1, 42, -56, 7, 14, -2, -28, 21, -3, 14, -4, -5, 1

Offset: 0

Peter Luschny, Mar 08 2016

The triangle of coefficients of the partial P-polynomials is A269941. For the definition of the inverse partial P-polynomials see the link 'P-transform'.

			[[1]],
[[0], [-1]],
[[0], [-1], [1]],
[[0], [-2, 1], [2], [-1]],
[[0], [-5, 5, -1], [5, -2], [-3], [1]],
[[0], [-14, 21, -3, -6, 1], [14, -12, 2], [-9, 3], [4], [-1]],
[[0], [-42,84,-28,-28,7,7,-1],[42,-56,7,14,-2],[-28,21,-3],[14,-4],[-5],[1]]
Replacing the sublists by their sums reduces the triangle to a signed version of the triangle A097805. The column 1 of sublists is A111785 in a different order.

Peter Luschny, The P-transform.

Cf. A097805, A111785, A268441, A268442, A269941.

# For function PMultiCoefficients see A269941.
PMultiCoefficients(7, inverse = True)

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

Petros Hadjicostas, May 28 2020

"Standard order" means as produced by Maple's "sort" command.
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.
The number of terms in the n-th row is A000041(n), while the sum of the terms is A000110(n).
The function Bell(n,k) in the PARI program below is a modification of a similar function in the PARI help files and uses the Faà di Bruno formula (cf. A036040).

			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.

E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.
E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
Peter Luschny, The Bell transform.

For different versions, see A178867 and A268441.

Cf. A000041, A000110, A036040.

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), "]", ","))

B_n(x[1], ..., x[n]) = Sum_{k=1..n} B_{n,k}(x[1], ..., x[n-k+1]), where B_{n,k} = B_{n,k}(x[1], ..., x[n-k+1]) are the partial exponential Bell polynomials that satisfy B_{n,1} = x[n] for n >= 1 and B_{n,k} = (1/k)*Sum_{m=k-1..n-1} binomial(n,m)*x[n-m]*B_{m,k-1} for n >= 2 and k = 2..n.

E.g.f.: Exp(Sum_{i >= 1} x_i*t^i/i!) = 1 + Sum_{n >= 1} B_n(x_1, x_2, ..., x_n)*t^n/n! [Comtet, p. 134, Eq. [3b]].

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A269941 Triangle read by rows, the coefficients of the partial P-polynomials.

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A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.

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A353131 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)!, n>=1, 1<=k<=n.

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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.

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A269942 Triangle read by rows, the coefficients of the inverse partial P-polynomials.

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Programs

Sage

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.

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