A305106 - OEIS

%I #11 Dec 14 2020 07:49:58

%S 1,1,2,4,7,12,21,34,55,87,138,211,324,486,727,1079,1584,2305,3337,

%T 4789,6830,9712,13689,19225,26841,37322,51598,71108,97580,133350,

%U 181558,246335,332991,448706,602607,806732,1077333,1433885,1903682,2520246,3328549,4383929

%N Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

%H Alois P. Heinz, <a href="/A305106/b305106.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(6) = 21 unitary factorizations:

%e (13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)

%e (2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)

%e (2*3*5)

%e The a(6) = 21 multiset partitions:

%e {{6}}

%e {{2,4}}

%e {{1,5}}

%e {{3,3}}

%e {{2,2,2}}

%e {{1,1,4}}

%e {{1,2,3}}

%e {{1,1,2,2}}

%e {{1,1,1,3}}

%e {{1,1,1,1,2}}

%e {{1,1,1,1,1,1}}

%e {{1},{5}}

%e {{1},{2,3}}

%e {{2},{4}}

%e {{2},{1,3}}

%e {{2},{1,1,1,1}}

%e {{1,1},{4}}

%e {{1,1},{2,2}}

%e {{3},{1,2}}

%e {{3},{1,1,1}}

%e {{1},{2},{3}}

%t Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]

%t (* Second program: *)

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];

%t T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;

%t a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ in A321878 *)

%Y Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.

%Y Row sums of A321878.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 25 2018