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

Author

Alois P. Heinz, Jul 21 2021

Keywords

• nonn
• tabl

Comments

Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i) into distinct factors; A(2,2) = 5: 2*3*6, 4*9, 3*12, 2*18, 36.

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

Columns (or rows) k=0-10 give: A000110, A087648, A322773, A322774, A346897, A346898, A346899, A346900, A346901, A346902, A346903.

Main diagonal gives A094574.

First upper (or lower) diagonal gives A322771.

Second upper (or lower) diagonal gives A322772.

Antidiagonal sums give A346518.

Cf. A002110, A045778, A322770, A346500.

Programs

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 *)

Formula

A(n,k) = A045778(A002110(n)*A002110(k)).

A(n,k) = A(k,n).

A(n,k) = A322770(abs(n-k),min(n,k)).