A370365
Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j having at least one set of consecutive numbers whose maximum (if j>0) is a multiple of j.
Original entry on oeis.org
0, 1, 2, 3, 4, 11, 77, 1571, 101924, 21824842, 18998281193, 63437859518312, 1037654210033812290, 72422876152852051595343, 27306605231809196751929593081, 50723306700937648229840111395656830, 510196838745355443955126736574361550469276
Offset: 0
-
b:= proc(n, k) option remember; `if`(k=0, signum(n), add(
(-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1))
end:
a:= n-> add(b(j, n-j), j=0..n):
seq(a(n), n=0..16);
A370368
Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j having no set of consecutive numbers whose maximum (if j>0) is a multiple of j.
Original entry on oeis.org
1, 1, 1, 1, 3, 18, 347, 20679, 4064088, 3206794270, 9817417580226, 147957639234186793, 9515125170594095021483, 3369265619091187775505912588, 5792039079391869138256364232105952, 55416702792637442337898498177490975722265
Offset: 0
-
b:= proc(n, k) `if`(k=0, `if`(n=0, 1, 0), add(
(-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
end:
a:= n-> add(b(j, n-j), j=0..n):
seq(a(n), n=0..15);
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.
Original entry on oeis.org
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
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;
- 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.
-
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
-
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 *)
-
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)])
Showing 1-3 of 3 results.
Comments