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A361948
Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} binomial((j + 1)*n - 1, n - 1) if n >= 1, and A(0, k) = 1 for all k.
2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 10, 15, 1, 1, 1, 1, 35, 280, 105, 1, 1, 1, 1, 126, 5775, 15400, 945, 1, 1, 1, 1, 462, 126126, 2627625, 1401400, 10395, 1, 1, 1, 1, 1716, 2858856, 488864376, 2546168625, 190590400, 135135, 1, 1
OFFSET
0,13
COMMENTS
Row n gives the leading coefficients of the set partition polynomials of type n. The sequence of these polynomial sequences starts: A097805, A048993, A156289, A291451, A291452, ...
LINKS
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Antoine Chambert-Loir, Combinatorics of partitions, blog post 2024.
Alexander Karpov, Generalized knockout tournament seedings, International Journal of Computer Science in Sport, vol. 17(2), 2018.
FORMULA
A(n, k) = (1/k!) * [x^k] P(n, k), where P(n, k) = k!*x^k if n = 0 and otherwise 1 if k = 0 and otherwise Sum_{j=1..k} binomial(n*k, n*j)*P(n, k-j)*x.
A(n, k) = (n*k)!*[x^(n*k)] exp(x^n/n!) for n >= 1. - Peter Luschny, Aug 15 2024
EXAMPLE
Array A(n, k) starts:
[0] 1, 1, 1, 1, 1, 1, ...
[1] 1, 1, 1, 1, 1, 1, ...
[2] 1, 1, 3, 15, 105, 945, ... A001147
[3] 1, 1, 10, 280, 15400, 1401400, ... A025035
[4] 1, 1, 35, 5775, 2627625, 2546168625, ... A025036
[5] 1, 1, 126, 126126, 488864376, 5194672859376, ... A025037
[6] 1, 1, 462, 2858856, 96197645544, 11423951396577720, ... A025038
.
Triangle A(n-k, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 1, 1;
[3] 1, 1, 1, 1;
[4] 1, 1, 3, 1, 1;
[5] 1, 1, 10, 15, 1, 1;
[6] 1, 1, 35, 280, 105, 1, 1;
MAPLE
A := (n, k) -> mul(binomial((j + 1)*n - 1, n - 1), j = 0..k-1):
seq(seq(A(n-k, k), k = 0..n), n = 0..9);
# Alternative: using recursion:
A := proc(n, k) local P; P := proc(n, k) option remember;
if n = 0 then return x^k*k! fi; if k = 0 then 1 else add(binomial(n*k, n*j)*
P(n, k-j)*x, j=1..k) fi end: coeff(P(n, k), x, k) / k! end:
seq(print(seq(A(n, k), k = 0..5)), n = 0..6);
# Alternative: using exponential generating function:
egf := n -> ifelse(n=0, 1, exp(x^n/n!)): ser := n -> series(egf(n), x, 8*n):
row := n -> local k; seq((n*k)!*coeff(ser(n), x, n*k), k = 0..6):
for n from 0 to 6 do [n], row(n) od; # Peter Luschny, Aug 15 2024
MATHEMATICA
A[n_, k_] := Product[Binomial[n (j + 1) - 1, n - 1], {j, 0, k - 1}]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 13 2023 *)
PROG
(SageMath)
def Arow(n, size):
if n == 0: return [1] * size
return [prod(binomial((j + 1)*n - 1, n - 1) for j in range(k)) for k in range(size)]
for n in range(7): print(Arow(n, 7))
# Alternative: using exponential generating function:
def SetPolyLeadCoeff(m, n):
x, z = var("x, z")
if m == 0: return 1
w = exp(2 * pi * I / m)
o = sum(exp(z * w ** k) for k in range(m)) / m
t = exp(x * (o - 1)).taylor(z, 0, m*n)
p = factorial(m*n) * t.coefficient(z, m*n)
return p.leading_coefficient(x)
for m in range(7):
print([SetPolyLeadCoeff(m, k) for k in range(6)])
CROSSREFS
Cf. A060540 (subarray), A370407 (antidiagonal sums, row sums).
Cf. A001147 (row 2), A025035 (row 3), A025036 (row 4), A025037 (row 5), A025038 (row 6), A025039 (row 7), A025040 (row 8), A025041 (row 9).
Cf. A088218 (column 2), A060542 (column 3), A082368 (column 4), A322252 (column 5), A057599 (main diagonal).
Sequence in context: A360161 A141901 A200473 * A180172 A327372 A374985
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Apr 13 2023
STATUS
approved