A322772 -id:A322772 - cp's OEIS frontend

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

Alois P. Heinz, Jul 21 2021

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.

			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, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

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.

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 *)

A(n,k) = A045778(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322770(abs(n-k),min(n,k)).

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

N. J. A. Sloane, Dec 30 2018

			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.)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
D. E. Knuth, Partitioning a multiset into submultisets, Email to N. J. A. Sloane, Dec 29 2018.

Rows include A094574, A322771, A322772.
Columns include A000110, A087648, A322773, A322774.
Main diagonal is A322775.

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 *)

Knuth gives a recurrence using the Bell numbers A000110 (see Maple program).

cp's OEIS Frontend

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.

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