A346517
Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k} into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 5, 3, 3, 5, 15, 9, 5, 9, 15, 52, 31, 18, 18, 31, 52, 203, 120, 70, 40, 70, 120, 203, 877, 514, 299, 172, 172, 299, 514, 877, 4140, 2407, 1393, 801, 457, 801, 1393, 2407, 4140, 21147, 12205, 7023, 4025, 2295, 2295, 4025, 7023, 12205, 21147
Offset: 0
A(2,2) = 5: 1122, 11|22, 1|122, 112|2, 1|12|2.
Square array A(n,k) begins:
1, 1, 2, 5, 15, 52, 203, 877, ...
1, 1, 3, 9, 31, 120, 514, 2407, ...
2, 3, 5, 18, 70, 299, 1393, 7023, ...
5, 9, 18, 40, 172, 801, 4025, 21709, ...
15, 31, 70, 172, 457, 2295, 12347, 70843, ...
52, 120, 299, 801, 2295, 6995, 40043, 243235, ...
203, 514, 1393, 4025, 12347, 40043, 136771, 875936, ...
877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, ...
...
Columns (or rows) k=0-10 give:
A000110,
A087648,
A322773,
A322774,
A346897,
A346898,
A346899,
A346900,
A346901,
A346902,
A346903.
First upper (or lower) diagonal gives
A322771.
Second upper (or lower) diagonal gives
A322772.
-
g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
g(n/d, d-1)), d=divisors(n) minus {1, n}))
end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n-1, j-1), j=1..n))
end:
A:= proc(n, k) option remember; `if`(n
-
(* Q is A322770 *)
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
A[n_, k_] := Q[Abs[n-k], Min[n, k]];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Aug 19 2021 *)
A322770
Array read by upwards antidiagonals: T(m,n) = number of set partitions into distinct parts of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 5, 9, 18, 40, 15, 31, 70, 172, 457, 52, 120, 299, 801, 2295, 6995, 203, 514, 1393, 4025, 12347, 40043, 136771, 877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, 4140, 12205, 38043, 124997, 431636, 1562071, 5908978, 23308546, 95668354
Offset: 0
The array begins:
1, 1, 5, 40, 457, 6995, 136771, ...
1, 3, 18, 172, 2295, 40043, 875936, ...
2, 9, 70, 801, 12347, 243235, 5908978, ...
5, 31, 299, 4025, 70843, 1562071, 41862462, ...
15, 120, 1393, 21709, 431636, 10569612, 310606617, ...
52, 514, 7023, 124997, 2781372, 75114998, 2407527172, ...
203, 2407, 38043, 764538, 18885177, 559057663, 19449364539, ...
...
- D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. (Background information.)
-
B := n -> combinat[bell](n):
Q := proc(m,n) local k; global B; option remember;
if n = 0 then B(m) else
(1/2)*( Q(m+2,n-1) + Q(m+1,n-1) - add( binomial(n-1,k)*Q(m,k), k=0..n-1) ); fi; end; # Q(m,n) (which is Knuth's notation) is T(m,n)
-
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
Table[Q[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)
Original entry on oeis.org
66, 249, 1075, 5133, 26683, 149410, 894124, 5683643, 38186471, 270054693, 2003233849, 15539557832, 125727364954, 1058517565329, 9254401000167, 83864649249661, 786443164853871, 7620060523287298, 76182494678422156, 784894262599538443, 8323844770001318027, 90767456208155581709
Offset: 0
- D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.
-
T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2);
vector(20, n, T(n-1, 3)) \\ Seiichi Manyama, Nov 21 2020
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