A188431 -id:A188431 - cp's OEIS frontend

A365922 Number of non-subset-sums of strict integer partitions of n.

0, 1, 2, 4, 8, 11, 18, 25, 38, 51, 70, 93, 122, 159, 206, 263, 328, 420, 514, 645, 776, 967, 1154, 1413, 1686, 2042, 2414, 2890, 3394, 4062, 4732, 5598, 6494, 7652, 8836, 10329, 11884, 13833, 15830, 18376, 20936, 24131, 27476, 31547, 35780, 40966, 46292, 52737

Offset: 1

Gus Wiseman, Sep 23 2023

nonn

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The a(6) = 11 ways, showing each strict partition and its non-subset-sums:
    (6): 1,2,3,4,5
   (51): 2,3,4
   (42): 1,3,5
  (321):

The complement (positive subset-sums) is A284640, non-strict A276024.
Weighted row sums of A365545, non-strict A365923.
Row sums of A365663, non-strict A046663.
The non-strict version is A365918.
The zero-full complement (subset-sums) is A365925, non-strict A304792.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A325799, A364272, A365658, A365919, A365921.

Table[Total[Length[Complement[Range[n], Total/@Subsets[#]]]& /@ Select[IntegerPartitions[n], UnsameQ@@#&]],{n,30}]

A325788 Number of complete strict necklace compositions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 20, 6, 16, 12, 10, 0, 84, 40, 74, 42, 66, 38, 22, 254, 238, 188, 356, 242, 272, 150, 148, 1140, 1058, 1208, 1546, 1288

Offset: 1

Gus Wiseman, May 22 2019

A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part (counted by A032153). A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

			The a(1) = 1 through a(16) = 6 complete strict necklace compositions (empty columns not shown):
  (1)  (12)  (123)  (124)  (1234)  (1253)  (1245)  (1264)  (12345)  (12634)
             (132)  (142)  (1324)  (1325)  (1326)  (1327)  (12354)  (13624)
                           (1423)  (1352)  (1542)  (1462)  (12435)  (14263)
                           (1432)  (1523)  (1623)  (1723)  (12453)  (14326)
                                                           (12543)  (14362)
                                                           (13254)  (16234)
                                                           (13425)
                                                           (13452)
                                                           (13524)
                                                           (13542)
                                                           (14235)
                                                           (14253)
                                                           (14325)
                                                           (14523)
                                                           (14532)
                                                           (15234)
                                                           (15243)
                                                           (15324)
                                                           (15342)
                                                           (15432)

Cf. A000740, A002033, A008965, A032153, A103295, A126796, A188431, A325684, A325785, A325786, A325787, A325790, A325791.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,30}]

A365832 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 0, 1, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 0, 4, 0, 0, 2, 2, 0, 0, 1, 0, 1, 0, 5, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 5, 0, 0, 2, 5, 0, 0, 0, 0, 2

Offset: 0

Gus Wiseman, Sep 28 2023

			The partition (7,6,1) has sums 1, 6, 7, 8, 13, 14, so is counted under T(14,6).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  0  1
  0  1  0  1  0
  0  1  0  2  0  0
  0  1  0  2  0  0  1
  0  1  0  3  0  0  0  1
  0  1  0  3  0  0  1  1  0
  0  1  0  4  0  0  0  3  0  0
  0  1  0  4  0  0  2  2  0  0  1
  0  1  0  5  0  0  0  5  0  0  0  1
  0  1  0  5  0  0  2  5  0  0  0  0  2
  0  1  0  6  0  0  0  8  0  0  0  1  0  2
  0  1  0  6  0  0  3  7  0  0  0  0  3  1  1
  0  1  0  7  0  0  0 12  0  0  0  1  0  4  0  2
  0  1  0  7  0  0  3 11  0  0  0  1  3  2  2  1  1
  0  1  0  8  0  0  0 16  0  0  0  1  0  7  0  3  0  2
  0  1  0  8  0  0  4 15  0  0  0  1  3  3  6  2  0  0  3
  0  1  0  9  0  0  0 21  0  0  0  2  0  9  0  7  0  1  0  4
  0  1  0  9  0  0  4 20  0  0  1  0  4  8  5  5  0  0  2  0  5
Row n = 14 counts the following partitions (A..E = 10..14):
  (E)  .  (D1)  .  .  (761)  (B21)  .  .  .  .  (6521)  (8321)  (7421)
          (C2)        (752)  (A31)              (6431)
          (B3)        (743)  (941)              (5432)
          (A4)               (932)
          (95)               (851)
          (86)               (842)
                             (653)

Row sums are A000009.
Rightmost column n = k is A188431, non-strict A126796.
The one-based weighted row sums are A284640.
The corresponding rank statistic is A299701.
The non-strict version is A365658.
Central column n = 2k in the non-strict case is A365660.
Reverse-weighted row-sums are A365922, non-strict A276024.
A000041 counts integer partitions.
A000124 counts distinct sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A137719, A304792, A364916.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,0,15},{k,0,n}]

A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 13, 15, 19, 22, 27, 31, 38, 43, 51, 59, 70, 79, 94, 107, 124, 143, 165, 188, 218, 248, 283, 324, 369, 419, 476, 540, 610, 691, 778, 878, 987, 1111, 1244, 1399, 1563, 1750, 1954, 2184, 2432, 2714, 3016, 3358, 3730, 4143

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

			The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
  (6)    (7)    (8)      (9)      (10)     (11)     (12)       (13)
  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)    (7,5)      (7,6)
  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)    (7,4)    (8,4)      (8,5)
         (6,1)  (7,1)    (7,2)    (8,2)    (8,3)    (9,3)      (9,4)
                (4,3,1)  (8,1)    (9,1)    (9,2)    (10,2)     (10,3)
                         (4,3,2)  (5,3,2)  (10,1)   (11,1)     (11,2)
                         (5,3,1)  (5,4,1)  (5,4,2)  (5,4,3)    (12,1)
                                  (6,3,1)  (6,3,2)  (6,4,2)    (6,4,3)
                                           (6,4,1)  (6,5,1)    (6,5,2)
                                           (7,3,1)  (7,3,2)    (7,4,2)
                                                    (7,4,1)    (7,5,1)
                                                    (8,3,1)    (8,3,2)
                                                    (5,4,2,1)  (8,4,1)
                                                               (9,3,1)
                                                               (6,4,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399*
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A365924 counts incomplete partitions, strict A365831.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367403 counts partitions without covering semi-sums, strict A367411.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A126796, A188431, A238628, A237113, A363225, A364272, A365658, A365918, A365925, A367410.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,15}]

A325988 Number of covering (or complete) factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A072911 at a(64) = 5, A072911(64) = 4.

A covering factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of some submultiset of the factors.

			The a(64) = 5 factorizations:
  (2*2*2*2*2*2)
  (2*2*2*2*4)
  (2*2*2*8)
  (2*2*4*4)
  (2*4*8)
The a(96) = 4 factorizations:
  (2*2*2*2*2*3)
  (2*2*2*3*4)
  (2*2*3*8)
  (2*3*4*4)

Cf. A002033, A103295, A126796, A188431.

Cf. A325684, A325781, A325786, A325788, A325986, A325989, A325990.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Union[Times@@@Subsets[#]]==Divisors[n]&]],{n,100}]

a(2^n) = A126796(n).

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

Original entry on oeis.org

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset: 0

Gus Wiseman, Jun 11 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

The ordered version is A072639, for run-lengths A333629.
The version for run-lengths is A325278, firsts in A182850 or A323014.
The run-sum trajectory is the iteration of A353832.
The first length-k row of A353840 has index a(k).
Other sequences pertaining to this trajectory are A353841-A353846.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

A364406 Number of permutations of [n] such that the minimal element of each cycle is also its length.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 6, 6, 0, 0, 720, 2160, 9360, 19440, 30240, 3659040, 21772800, 228614400, 1632960000, 11125900800, 73025971200, 1708337433600, 15442053580800, 254260755302400, 3318429200486400, 46929444097536000, 546974781889536000, 7312714579602432000

Offset: 0

Alois P. Heinz, Jul 22 2023

nonn

			a(0) = 1: () the empty permutation.
a(1) = 1: (1).
a(3) = 1: (1)(23).
a(6) = 6: (1)(24)(356), (1)(24)(365), (1)(25)(346), (1)(25)(364),
  (1)(26)(345), (1)(26)(354).
a(7) = 6: (1)(23)(4567), (1)(23)(4576), (1)(23)(4657), (1)(23)(4675),
  (1)(23)(4756), (1)(23)(4765).

Alois P. Heinz, Table of n, a(n) for n = 0..474
Wikipedia, Permutation

Cf. A000142, A007837, A188431, A362362, A364207, A364277 A364279 A364281 A364283.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n+1, 0, b(n-i, i-1)*binomial(n-i, i-1)*(i-1)!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);

b[n_, i_] := b[n, i] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1] + If[2*i > n + 1, 0, b[n - i, i - 1]*Binomial[n - i, i - 1]*(i - 1)!]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)

A325986 Heinz numbers of complete strict integer partitions.

Original entry on oeis.org

1, 2, 6, 30, 42, 210, 330, 390, 462, 510, 546, 714, 798, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7854, 8778, 8970, 9282, 9570, 9690, 10230, 10374, 10626, 11310, 11730, 12090, 12210, 12558, 13398, 13566, 14322, 14430

Offset: 1

Gus Wiseman, May 30 2019

nonn

Strict partitions are counted by A000009, while complete partitions are counted by A126796.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is complete (A126796, A325781) if every number from 0 to n is the sum of some submultiset of the parts.
The enumeration of these partitions by sum is given by A188431.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     42: {1,2,4}
    210: {1,2,3,4}
    330: {1,2,3,5}
    390: {1,2,3,6}
    462: {1,2,4,5}
    510: {1,2,3,7}
    546: {1,2,4,6}
    714: {1,2,4,7}
    798: {1,2,4,8}
   2310: {1,2,3,4,5}
   2730: {1,2,3,4,6}
   3570: {1,2,3,4,7}
   3990: {1,2,3,4,8}
   4290: {1,2,3,5,6}
   4830: {1,2,3,4,9}
   5610: {1,2,3,5,7}

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A304793.

Cf. A325780, A325781, A325782, A325788, A325790, A325988.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Select[Range[1000],SquareFreeQ[#]&&Union[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Intersection of A005117 (strict partitions) and A325781 (complete partitions).

A326036 Number of uniform complete integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 2, 2, 2, 2, 6, 3, 3, 5, 5, 3, 8, 5, 11, 10, 10, 9, 19, 13, 15, 17, 21, 18, 35, 26, 39, 40, 50, 50, 77, 63, 84, 88, 113, 103, 146, 132, 171, 180, 212, 214, 292, 276, 345, 363, 435, 442, 561, 569, 694, 729, 853, 891, 1108

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition of n is uniform if all parts appear with the same multiplicity, and complete if every nonnegative integer up to n is the sum of some submultiset.

			The initial terms count the following partitions:
   0: ()
   1: (1)
   2: (11)
   3: (21)
   3: (111)
   4: (1111)
   5: (11111)
   6: (321)
   6: (2211)
   6: (111111)
   7: (421)
   7: (1111111)
   8: (3311)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (4321)
  10: (1111111111)
  11: (5321)
  11: (11111111111)

Cf. A002033, A047966, A072774, A108917, A126796, A188431, A276024.

Cf. A325781, A325791, A325988, A326020, A326035, A326037.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#]&&Sort[sums[Sort[#]]]==Range[Total[#]]&]],{n,0,30}]

A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.

Original entry on oeis.org

1, 3, 9, 21, 22, 27, 63, 76, 81, 117, 147, 175, 186, 189, 243, 248, 273, 286, 290, 322, 345, 351, 399, 418, 441, 513, 516, 567, 688, 715, 729, 819, 1029, 1053, 1062, 1156, 1180, 1197, 1323, 1375, 1416, 1484, 1521, 1539, 1701, 1827, 1888, 1911, 2068, 2115, 2130

Offset: 1

Gus Wiseman, Sep 25 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
     1: {}
     3: {2}
     9: {2,2}
    21: {2,4}
    22: {1,5}
    27: {2,2,2}
    63: {2,2,4}
    76: {1,1,8}
    81: {2,2,2,2}
   117: {2,2,6}
   147: {2,4,4}
   175: {3,3,4}
   186: {1,2,11}
   189: {2,2,2,4}
   243: {2,2,2,2,2}

The LHS is A304793, counted by A365658, with empty sets A299701.
The RHS is A325799, counted by A365923 (strict A365545).
A046663 counts partitions without a subset summing to k, strict A365663.
A056239 adds up prime indices, row sums of A112798.
A276024 counts positive subset-sums of partitions, strict A284640.
A325781 ranks complete partitions, counted by A126796.
A365830 ranks incomplete partitions, counted by A365924.
A365918 counts non-subset-sums of partitions, strict A365922.
Cf. A001223, A005117, A006827, A073491, A188431, A304792, A365831.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
smu[y_]:=Union[Total/@Rest[Subsets[y]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Select[Range[100],Length[smu[prix[#]]]==Length[nmz[prix[#]]]&]

Positive integers k such that A304793(k) = A325799(k).

cp's OEIS Frontend

A365922 Number of non-subset-sums of strict integer partitions of n.

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A325788 Number of complete strict necklace compositions of n.

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A365832 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.

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A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

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A325988 Number of covering (or complete) factorizations of n.

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A353743 Least number with run-sum trajectory of length k; a(0) = 1.

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A364406 Number of permutations of [n] such that the minimal element of each cycle is also its length.

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Maple

Mathematica

A325986 Heinz numbers of complete strict integer partitions.

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Mathematica

Formula

A326036 Number of uniform complete integer partitions of n.

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