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.
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
Examples
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, ...
...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
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 -
Mathematica
(* 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 *)
Comments