A325098 Number of binary carry-connected integer partitions of n.

1, 1, 2, 2, 4, 4, 7, 7, 13, 15, 23, 27, 42, 50, 72, 88, 125, 153, 211, 258, 349, 430, 569, 698, 914, 1119, 1444, 1765, 2252, 2745, 3470, 4214, 5276, 6387, 7934, 9568, 11800, 14181, 17379, 20818, 25351, 30264, 36668, 43633, 52589, 62394, 74872, 88576, 105818

Offset

0,3

Comments

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Example

The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (322)      (44)
                    (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (222)     (511)      (62)
                                     (321)     (3211)     (71)
                                     (3111)    (31111)    (332)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)

Maple

h:= proc(n, s) local i, m; m:= n;
      for i in s do m:= Bits[Or](m, i) od; {m}
    end:
g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
              h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
b:= proc(n, i, s) option remember; `if`(n=0, `if`(nops(s)>1, 0, 1),
      `if`(i<1, 0, b(n, i-1, s)+ b(n-i, min(i, n-i), g(i, s))))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 29 2019

Mathematica

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], Length[csm[binpos/@#]]<=1&]], {n, 0, 20}]
(* Second program: *)
h[n_, s_] := Module[{i, m = n}, Do[m = BitOr[m, i], {i, s}]; {m}];
g[n_, s_] := Function[w, If[w == {}, s ~Union~ {n}, (s ~Complement~ w) ~Union~
    h[n, w]]][Select[s, BitAnd[n, #] > 0&]];
b[n_, i_, s_] := b[n, i, s] = If[n == 0, If[Length[s] > 1, 0, 1],
    If[i < 1, 0, b[n, i - 1, s] + b[n - i, Min[i, n - i], g[i, s]]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 50] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)

Crossrefs

Cf. A050315, A080572, A247935, A267610, A267700.

Cf. A325096, A325099, A325104, A325106, A325108, A325110, A325118, A325119.

Sequence in context: A131524 A089075 A355195 * A368688 A208963 A011142

Adjacent sequences: A325095 A325096 A325097 * A325099 A325100 A325101

Keyword

nonn

Author

Gus Wiseman, Mar 28 2019

Extensions

a(21)-a(48) from Alois P. Heinz, Mar 29 2019

Status

approved