A050315 -id:A050315 - cp's OEIS frontend

A336137 Number of set partitions of the binary indices of n with equal block-sums.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(n) set partitions for n = 7, 59, 119, 367, 127:
  {123}    {12456}      {123567}      {1234679}    {1234567}
  {12}{3}  {126}{45}    {1236}{57}    {12346}{79}  {1247}{356}
           {15}{24}{6}  {156}{237}    {1249}{367}  {1256}{347}
                        {17}{26}{35}  {1267}{349}  {1346}{257}
                                      {169}{2347}  {167}{2345}
                                                   {16}{25}{34}{7}
The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
  {{2,7},{3,6},{4,5},{9}}
  {{2,4,6},{3,9},{5,7}}
  {{2,7,9},{3,4,5,6}}
  {{2,3,4,9},{5,6,7}}
  {{2,3,6,7},{4,5,9}}
  {{2,4,5,7},{3,6,9}}
  {{2,3,4,5,6,7,9}}
so a(382) = 7.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

These set partitions are counted by A035470.
The version for twice-partitions is A279787.
The version for partitions of partitions is A305551.
The version for factorizations is A321455.
The version for normal multiset partitions is A326518.
The version for distinct block-sums is A336138.
Set partitions of binary indices are A050315.
Normal multiset partitions with equal lengths are A317583.
Normal multiset partitions with equal averages are A326520.
Multiset partitions with equal block-sums are ranked by A326534.
Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]

A325102 Number of ordered pairs of positive integers up to n with no binary carries.

Original entry on oeis.org

0, 0, 2, 2, 8, 10, 12, 12, 26, 32, 38, 40, 46, 48, 50, 50, 80, 94, 108, 114, 128, 134, 140, 142, 156, 162, 168, 170, 176, 178, 180, 180, 242, 272, 302, 316, 346, 360, 374, 380, 410, 424, 438, 444, 458, 464, 470, 472, 502, 516, 530, 536, 550, 556, 562, 564, 578

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

			The a(2) = 2 through a(6) = 12 pairs:
  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)
  (2,1)  (2,1)  (1,4)  (1,4)  (1,4)  (1,4)
                (2,1)  (2,1)  (1,6)  (1,6)
                (2,4)  (2,4)  (2,1)  (2,1)
                (3,4)  (2,5)  (2,4)  (2,4)
                (4,1)  (3,4)  (2,5)  (2,5)
                (4,2)  (4,1)  (3,4)  (3,4)
                (4,3)  (4,2)  (4,1)  (4,1)
                       (4,3)  (4,2)  (4,2)
                       (5,2)  (4,3)  (4,3)
                              (5,2)  (5,2)
                              (6,1)  (6,1)

Cf. A006218, A050315, A080572, A247935, A267610, A267700.

Cf. A325094, A325096, A325098, A325103, A325104, A325106, A325108, A325123.

Table[Length[Select[Tuples[Range[n],2],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,30}]

a(n) = 2 * A325103(n).

A325123 Number of divisible pairs of positive integers up to n with no binary carries.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 4, 4, 7, 7, 9, 9, 12, 12, 13, 13, 17, 17, 19, 19, 22, 22, 23, 23, 28, 28, 29, 29, 31, 31, 32, 32, 37, 37, 39, 39, 44, 44, 45, 45, 50, 50, 52, 52, 54, 54, 55, 55, 62, 62, 64, 64, 66, 66, 68, 68, 72, 72, 73, 73, 76, 76, 77, 77, 83, 83, 85, 85

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.

a(2k+1) = a(2k), since an odd number and any divisor will overlap in the last digit. Additionally, a(2k+2) > a(2k+1) because the pair {1,2k+2} is always valid. Therefore, every term appears exactly twice. - Charlie Neder, Apr 02 2019

			The a(2) = 1 through a(11) = 9 pairs:
  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}   {1,2}
                {1,4}  {1,4}  {1,4}  {1,4}  {1,4}  {1,4}  {1,4}   {1,4}
                {2,4}  {2,4}  {1,6}  {1,6}  {1,6}  {1,6}  {1,6}   {1,6}
                              {2,4}  {2,4}  {1,8}  {1,8}  {1,8}   {1,8}
                                            {2,4}  {2,4}  {2,4}   {2,4}
                                            {2,8}  {2,8}  {2,8}   {2,8}
                                            {4,8}  {4,8}  {4,8}   {4,8}
                                                          {1,10}  {1,10}
                                                          {5,10}  {5,10}

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325124.

Table[Length[Select[Tuples[Range[n],2],Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,20}]

A325099 Number of binary carry-connected strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 1, 4, 5, 8, 6, 11, 11, 15, 13, 18, 20, 30, 29, 43, 49, 68, 66, 84, 94, 125, 131, 165, 184, 237, 251, 291, 315, 383, 408, 486, 536, 663, 714, 832, 912, 1104, 1195, 1405, 1554, 1877, 2046, 2348, 2559, 2998, 3256, 3730, 4084, 4793, 5230, 5938

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

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.

			The a(1) = 1 through a(11) = 6 strict partitions (A = 10, B = 11):
  (1)  (2)  (3)  (4)   (5)   (6)    (7)  (8)   (9)    (A)    (B)
                 (31)  (32)  (51)        (53)  (54)   (64)   (65)
                             (321)       (62)  (63)   (73)   (74)
                                         (71)  (72)   (91)   (632)
                                               (531)  (532)  (731)
                                                      (541)  (5321)
                                                      (631)
                                                      (721)

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

Cf. A325095, A325096, A325098, A325104, A325106, A325110, A325119.

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],UnsameQ@@#&&Length[csm[binpos/@#]]<=1&]],{n,0,30}]

A325100 Heinz numbers of strict integer partitions with no binary carries.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 41, 42, 43, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 74, 79, 83, 86, 89, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 122, 123, 127, 131, 133, 137, 139, 142, 149, 151, 157, 158, 159

Offset: 1

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices have no carries. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  41: {13}
  42: {1,2,4}

Cf. A050315, A056239, A080572, A112798, A247935, A267610.

Cf. A325095, A325096, A325097, A325100, A325101, A325103, A325110, A325119.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],SquareFreeQ[#]&&stableQ[PrimePi/@First/@FactorInteger[#],Intersection[binpos[#1],binpos[#2]]!={}&]&]

A325124 Number of divisible pairs of positive integers up to n with at least one binary carry.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 10, 12, 13, 16, 18, 20, 23, 25, 28, 32, 33, 35, 39, 41, 44, 48, 51, 53, 56, 59, 62, 66, 70, 72, 79, 81, 82, 86, 88, 92, 96, 98, 101, 105, 108, 110, 116, 118, 122, 128, 131, 133, 136, 139, 143, 147, 151, 153, 159, 163, 167, 171, 174, 176, 185

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.

			The a(1) = 1 through a(8) = 13 pairs:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)  (1,5)  (1,5)  (1,5)
                (3,3)  (3,3)  (2,2)  (2,2)  (1,7)  (1,7)
                       (4,4)  (3,3)  (2,6)  (2,2)  (2,2)
                              (4,4)  (3,3)  (2,6)  (2,6)
                              (5,5)  (3,6)  (3,3)  (3,3)
                                     (4,4)  (3,6)  (3,6)
                                     (5,5)  (4,4)  (4,4)
                                     (6,6)  (5,5)  (5,5)
                                            (6,6)  (6,6)
                                            (7,7)  (7,7)
                                                   (8,8)

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610, A307230.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325123.

Table[Length[Select[Tuples[Range[n],2],Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]!={}&]],{n,0,20}]

a(n) = A307230(n) + n.

A088512 Number of partitions of n into two parts whose xor-sum is n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 0, 1, 1, 3, 1, 3, 3, 7, 0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7

Offset: 0

Naohiro Nomoto, Nov 14 2003

			G.f. = x^3 + x^5 + x^6 + 3*x^7 + x^9 + x^10 + 3*x^11 + x^12 + 3*x^13 + 3*x^14 + ...
From _Emmanuele Villa_, Nov 19 2016: (Start)
For n = 47, the highest power of 2 less than n is 32, so a(47) = A001316(47-32) - 1 = A001316(15) - 1 = 16 - 1 = 15.
For n = 63, the highest power of 2 less than n is 32, so a(63) = A001316(63-32) - 1 = A001316(31) - 1 = 32 - 1 = 31. (End)

Cf. A050315.

Table[2^DigitCount[# - 2^(Floor@ Log2@ # - Boole@ IntegerQ@ Log2@ #) - 1 + Boole[# == 1]/2, 2, 1] - 1 &[n + 1], {n, 0, 72}] (* Michael De Vlieger, Nov 18 2016 *)
a[ n_] := Which[ n < 3, 0, EvenQ[n], a @ Quotient[n, 2], True, a[ Quotient[n, 2]] 2 + 1]; (* Michael Somos, Dec 04 2016 *)

a(n) = sum(m=1, n\2, bitxor(m,n-m)==n); \\ Michel Marcus, Dec 03 2016

{a(n) = if( n<3, 0, n%2, a(n\2)*2 + 1, a(n\2))}; /* Michael Somos, Dec 04 2016 */

a(0) = 0, a(n) = A001316(n-m)-1, where m is the highest power of 2 less than n. - Emmanuele Villa, Nov 19 2016

a(2*n) = a(n), a(2*n + 1) = 2*a(n) + 1. - Michael Somos, Dec 04 2016

A307431 Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 1, 0, 4, 0, 2, 0, 1, 1, 5, 0, 2, 2, 0, 1, 0, 7, 0, 2, 0, 5, 0, 1, 1, 8, 1, 2, 2, 6, 1, 0, 1, 0, 11, 0, 4, 0, 12, 0, 2, 0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2, 0, 1, 0, 15, 0, 5, 0, 28, 0, 2, 0, 5, 0, 1, 1, 17, 1, 5, 5, 35, 0, 2, 2, 6, 2

Offset: 0

Alois P. Heinz, Apr 08 2019

			T(6,1) = 1: 111111.
T(6,2) = 1: 222.
T(6,3) = 5: 11112, 1122, 1113, 123, 33.
T(6,5) = 2: 114, 15.
T(6,6) = 2: 24, 6.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0,  2;
  0, 1, 1,  2, 1;
  0, 1, 0,  4, 0, 2;
  0, 1, 1,  5, 0, 2, 2;
  0, 1, 0,  7, 0, 2, 0,  5;
  0, 1, 1,  8, 1, 2, 2,  6, 1;
  0, 1, 0, 11, 0, 4, 0, 12, 0, 2;
  0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Bitwise operation
Wikipedia, Partition (number theory)

Columns k=0-1 give: A000007, A057427.
Row sums give: A000041.
Main diagonal gives A050315.
Cf. A050314 (the same for XOR), A307432 (the same for AND).

b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), Bits[Or](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);

A307505 Number T(n,k) of partitions of n into distinct parts whose bitwise XOR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 0, 0, 0, 5, 1, 0, 5, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2

Offset: 0

Alois P. Heinz, Apr 11 2019

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 2;
  0, 0, 1, 0, 1;
  0, 1, 0, 0, 0, 2;
  1, 0, 0, 0, 1, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 5;
  0, 0, 0, 0, 1, 0, 4, 0, 1;
  0, 1, 0, 0, 0, 4, 0, 1, 0, 2;
  1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2;
  ...

Alois P. Heinz, Rows n = 0..360, flattened
Wikipedia, Bitwise operation
Wikipedia, Partition (number theory)

Bisection (even part) of column k=0 gives A307506.
Row sums give A000009.
Main diagonal gives A050315.
Cf. A050314.

b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i-1), Bits[Xor](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);

T(n,k) = 0 if n+k is odd.

A336140 Number of ways to choose a set partition of the parts of a strict integer composition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 39, 43, 73, 107, 497, 531, 951, 1345, 2125, 8789, 9929, 16953, 24723, 38347, 52717, 219131, 240461, 419715, 600075, 938689, 1278409, 1928453, 6853853, 7815657, 13205247, 19051291, 29325121, 40353995, 60084905, 80722899, 277280079, 312239953

Offset: 0

Gus Wiseman, Jul 16 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Set partitions are A000110.
Strict compositions are A032020.
Set partitions of binary indices are A050315.
Set partitions of strict partitions are A294617.
Cf. A000009, A001055, A008289, A035470, A063834, A072574, A137341, A279375.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 30 2020

Table[Sum[BellB[Length[ctn]],{ctn,Join@@Permutations/@Select[ IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0,
     BellB[p]*p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n) = Sum_{k = 0..n} A000110(k) * A072574(n,k) = Sum_{k = 0..n} k! * A000110(k) * A008289(n,k).

cp's OEIS Frontend

A336137 Number of set partitions of the binary indices of n with equal block-sums.

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A325102 Number of ordered pairs of positive integers up to n with no binary carries.

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A325123 Number of divisible pairs of positive integers up to n with no binary carries.

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A325099 Number of binary carry-connected strict integer partitions of n.

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A325100 Heinz numbers of strict integer partitions with no binary carries.

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A325124 Number of divisible pairs of positive integers up to n with at least one binary carry.

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A088512 Number of partitions of n into two parts whose xor-sum is n.

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PARI

PARI

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A307431 Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

A307505 Number T(n,k) of partitions of n into distinct parts whose bitwise XOR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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