A304792 -id:A304792 - cp's OEIS frontend

A276024 Number of positive subset sums of integer partitions of n.

1, 3, 7, 14, 27, 47, 81, 130, 210, 319, 492, 718, 1063, 1512, 2178, 3012, 4237, 5765, 7930, 10613, 14364, 18936, 25259, 32938, 43302, 55862, 72694, 92797, 119499, 151468, 193052, 242748, 307135, 383315, 481301, 597252, 744199, 918030, 1137607, 1395101, 1718237, 2098096, 2569047, 3121825, 3805722

Offset: 1

Gus Wiseman, Aug 16 2016

nonn

For a multiset p of positive integers summing to n, a pair (t,p) is defined to be a positive subset sum if there exists a nonempty submultiset of p summing to t. Positive integers with positive subset sums form a multiorder. This sequence is dominated by A122768 (submultisets of integer partitions of n).

			The a(4)=14 positive subset sums are: {(4,4), (1,31), (3,31), (4,31), (2,22), (4,22), (1,211), (2,211), (3,211), (4,211), (1,1111), (2,1111), (3,1111), (4,1111)}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..150
Konstantinos Koiliaris and Chao Xu, A Faster Pseudopolynomial Time Algorithm for Subset Sum, arXiv:1507.02318 [cs.DS], 2015-2016.
Gus Wiseman, Comcategories and Multiorders (pdf version)

Cf. A122768, A063834, A262671.

sums[ptn_?OrderedQ]:=sums[ptn]=If[Length[ptn]===1,ptn,Module[{pri,sms},
pri=Union[Table[Delete[ptn,i],{i,Length[ptn]}]];
sms=Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@pri;
Union@@sms
]];
Table[Total[Length[sums[Sort[#]]]&/@IntegerPartitions[n]],{n,1,25}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0, Length[s], If[i < 1, 0, b[n, i - 1, s] + b[n - i, Min[n - i, i], {#, # + i}& /@ s // Flatten // Union]]];
a[n_] := b[n, n, {0}] - PartitionsP[n];
Array[a, 45] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz in A304792 *)

# uses A304792_T
from sympy import npartitions
def A276024(n): return A304792_T(n,n,(0,),1) - npartitions(n) # Chai Wah Wu, Sep 25 2023

A126796 Number of complete partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 31, 39, 55, 71, 100, 125, 173, 218, 291, 366, 483, 600, 784, 971, 1244, 1538, 1957, 2395, 3023, 3693, 4605, 5604, 6942, 8397, 10347, 12471, 15235, 18309, 22267, 26619, 32219, 38414, 46216, 54941, 65838, 77958, 93076, 109908

Offset: 0

Brian Hopkins, Feb 20 2007

A partition of n is complete if every number 1 to n can be represented as a sum of parts of the partition. This generalizes perfect partitions, where the representation for each number must be unique.
A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters, Mar 22 2007
For n > 0: a(n) = sum of n-th row in A261036 and also a(floor(n/2)) = A261036(n,floor((n+1)/2)). - Reinhard Zumkeller, Aug 08 2015
a(n+1) is the number of partitions of n such that each part is no more than 2 more than the sum of all smaller parts (generalizing Adams-Watters's criterion). Bijection: each partition counted by a(n+1) must contain a 1, removing that gives a desired partition of n. - Brian Hopkins, May 16 2017
A partition (x_1, ..., x_k) is complete if and only if 1, x_1, ..., x_k is a "regular sequence" (see A003513 for definition). As a result, the number of complete partitions with n parts is given by A003513(n+1). - Nathaniel Johnston, Jun 29 2023

			There are a(5) = 4 complete partitions of 5:
  [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], and [1, 1, 3].
G.f.: 1 = 1*(1-x) + 1*x*(1-x)*(1-x^2) + 1*x^2*(1-x)*(1-x^2)*(1-x^3) + 2*x^3*(1-x)*(1-x^2)*(1-x^3)*(1-x^4) + 2*x^4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5) + ...
From _Gus Wiseman_, Oct 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (11)  (21)   (211)   (221)    (321)     (421)      (3221)
             (111)  (1111)  (311)    (2211)    (2221)     (3311)
                            (2111)   (3111)    (3211)     (4211)
                            (11111)  (21111)   (4111)     (22211)
                                     (111111)  (22111)    (32111)
                                               (31111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 301 terms from David W. Wilson)
George E. Andrews, George Beck, and Brian Hopkins, On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions, Annals of Comb., 24(2020), pp. 217-224.
George Beck and Shane Chern, Reciprocity between partitions and compositions, arXiv:2108.04363 [math.CO], 2021.
Nathaniel Johnston and Sarah Plosker, Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs, arXiv:2308.15611 [math.CO], 2023.
SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

Cf. A002033, A003513, A209405, A261036, A286929, A286097.
For parts instead of sums we have A000009 (sc. coverings), ranks A055932.
The strict case is A188431, complement A365831.
These partitions have ranks A325781.
First column k = 0 of A365923.
The complement is counted by A365924, ranks A365830.
Cf. A000041, A018818, A046663, A047967, A276024, A304792, A325799, A365543, A365658, A365918, A365921.

import Data.MemoCombinators (memo3, integral, Memo)
a126796 n = a126796_list !! n
a126796_list = map (pMemo 1 1) [0..] where
   pMemo = memo3 integral integral integral p
   p   0 = 1
   p s k m
     | k > min m s = 0
     | otherwise   = pMemo (s + k) k (m - k) + pMemo s (k + 1) m
-- Reinhard Zumkeller, Aug 07 2015

isCompl := proc(p,n) local m,pers,reps,f,lst,s; reps := {}; pers := combinat[permute](p); for m from 1 to nops(pers) do lst := op(m,pers); for f from 1 to nops(lst) do s := add( op(i,lst),i=1..f); reps := reps union {s}; od; od; for m from 1 to n do if not m in reps then RETURN(false); fi; od; RETURN(true); end: A126796 := proc(n) local prts, res,p; prts := combinat[partition](n); res :=0; for p from 1 to nops(prts) do if isCompl(op(p,prts),n) then res := res+1; fi; od; RETURN(res); end: for n from 1 to 40 do printf("%d %d ",n,A126796(n)); od; # R. J. Mathar, Feb 27 2007
# At the beginning of the 2nd Maple program replace the current 15 by any other positive integer n in order to obtain a(n). - Emeric Deutsch, Mar 04 2007
with(combinat): a:=proc(n) local P,b,k,p,S,j: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i],i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: end: seq(a(n),n=1..30); # Emeric Deutsch, Mar 04 2007
with(combinat): n:=15: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i],i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: b; # Emeric Deutsch, Mar 04 2007

T[n_, k_] := T[n, k] = If[k <= 1, 1, If[n < 2k-1, T[n, Floor[(n+1)/2]], T[n, k-1] + T[n-k, k]]];
a[n_] := T[n, Floor[(n+1)/2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 11 2017, after Franklin T. Adams-Watters *)
nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]]; Table[Length[Select[IntegerPartitions[n], nmz[#]=={}&]],{n,0,15}] (* Gus Wiseman, Oct 14 2023 *)

{T(n,k)=if(k<=1,1,if(n<2*k-1,T(n,floor((n+1)/2)),T(n,k-1)+T(n-k,k)))}
{a(n)=T(n,floor((n+1)/2))} /* If modified to save earlier results, this would be efficient. */ /* Franklin T. Adams-Watters, Mar 22 2007 */

/* As coefficients in g.f.: */
{a(n)=local(A=[1,1]);for(i=1,n+1,A=concat(A,0);A[#A]=polcoeff(1-sum(m=1,#A,A[m]*x^m*prod(k=1,m,1-x^k +x*O(x^#A))),#A) );A[n+1]}
for(n=0,50,print1(a(n),",")) /* Paul D. Hanna, Mar 06 2012 */

G.f.: 1 = Sum_{n>=0} a(n)*x^n*Product_{k=1..n+1} (1-x^k). - Paul D. Hanna, Mar 08 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n) + (25/16 - 1679*Pi^2/6912)/n). - Vaclav Kotesovec, May 24 2018, extended Nov 02 2019
a(n) = A000041(n) - A365924(n). - Gus Wiseman, Oct 14 2023

More terms from R. J. Mathar, Feb 27 2007
More terms from Emeric Deutsch, Mar 04 2007
Further terms from Franklin T. Adams-Watters and David W. Wilson, Mar 22 2007

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 7, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 5, 7, 4, 8, 2, 6, 4, 7, 2, 8, 2, 4, 6, 6, 4, 8, 2, 8, 5, 4, 2, 9, 4, 4, 4

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer n is a subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Position of first appearance of n appears to be A259941(n-1) = least Heinz number of a complete partition of n-1. - Gus Wiseman, Nov 16 2023

			The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.
Positions of first appearances are A259941.
The triangle for this rank statistic is A365658.
The semi version is A366739, sum A366738, strict A366741.
Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.

Table[Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]

a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).

Comment corrected by Gus Wiseman, Aug 09 2024

A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

Original entry on oeis.org

1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304

Offset: 1

N. J. A. Sloane

			Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - _N. J. A. Sloane_, Jun 03 2012
From _Gus Wiseman_, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
  (11)  (22)    (33)      (44)
        (211)   (321)     (422)
        (1111)  (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.

Column m=2 of A213086.
Bisection of A276107.
Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.
The strict version is A237258, ranked by A357854.
Ranked by A357976 = positions of nonzero terms in A357879.
A122768 counts distinct submultisets of partitions.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A108917, A235130, A237194, A300061.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,
      b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,
      max(x, n-i-x)), `if`(xn, NULL, max(x-i, n-x))}[], s)))))
    end:
a:= n-> b(2*n, n, {n}):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

Mathematica

b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *) primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s]; Table[Length[Select[IntegerPartitions[2n],MemberQ[Total/@subptns[#],n]&]],{n,10}] (* Gus Wiseman, Oct 27 2022 *)

Python

from itertools import combinations_with_replacement from sympy.utilities.iterables import partitions def A002219(n): return len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023

Formula

See A213074 for Metropolis and Stein's formulas.

a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n).

a(n) = A276107(2*n). - Max Alekseyev, Oct 17 2022

Extensions

Better description from Vladeta Jovovic, Mar 06 2000

More terms from Christian G. Bower, Oct 12 2001

Edited by N. J. A. Sloane, Jun 03 2012

More terms from Alois P. Heinz, Jul 10 2012

A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 3, 3, 5, 7, 5, 5, 5, 5, 7, 11, 7, 8, 6, 8, 7, 11, 15, 11, 11, 11, 11, 11, 11, 15, 22, 15, 17, 15, 14, 15, 17, 15, 22, 30, 22, 23, 23, 22, 22, 23, 23, 22, 30, 42, 30, 33, 30, 33, 25, 33, 30, 33, 30, 42

Offset: 0

Gus Wiseman, Sep 16 2023

Rows are palindromic.

			Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   3   3   5
   7   5   5   5   5   7
  11   7   8   6   8   7  11
  15  11  11  11  11  11  11  15
  22  15  17  15  14  15  17  15  22
  30  22  23  23  22  22  23  23  22  30
  42  30  33  30  33  25  33  30  33  30  42
  56  42  45  44  44  43  43  44  44  45  42  56
  77  56  62  58  62  56  53  56  62  58  62  56  77
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (321)     (411)     (411)     (51)
  (42)      (321)     (321)     (3111)    (321)     (321)     (42)
  (411)     (3111)    (3111)    (2211)    (3111)    (3111)    (411)
  (33)      (2211)    (222)     (21111)   (222)     (2211)    (33)
  (321)     (21111)   (2211)    (111111)  (2211)    (21111)   (321)
  (3111)    (111111)  (21111)             (21111)   (111111)  (3111)
  (222)               (111111)            (111111)            (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Robert Price, Table of n, a(n) for n = 0..324

Columns k = 0 and k = n are A000041.
Central column n = 2k is A002219.
The complement is counted by A046663, strict A365663.
Row sums are A304792.
For subsets instead of partitions we have A365381.
The strict case is A365661.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A088809, A093971, A122768, A108917, A299701, A364911, A365541, A365658.

Table[Length[Select[IntegerPartitions[n],MemberQ[Total/@Subsets[#],k]&]],{n,0,15},{k,0,n}]

A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 5, 3, 4, 4, 4, 4, 4, 4, 4, 7, 5, 7, 8, 7, 5, 7, 8, 7, 7, 8, 8, 7, 7, 8, 12, 9, 12, 9, 17, 9, 12, 9, 12, 14, 11, 12, 12, 13, 13, 12, 12, 11, 14, 21, 15, 19, 15, 21, 24, 21, 15, 19, 15, 21, 24, 19, 20, 19, 21, 22, 22, 21, 19, 20, 19, 24, 34, 23, 30, 24, 30, 25, 46, 25, 30, 24, 30, 23, 34

Offset: 2

N. J. A. Sloane

			For n = 4 there are two partitions (4, 2+2) with no subsum equal to 1, two (4, 3+1) with no subsum equal to 2 and two (4, 2+2) with no subsum equal to 3.
Triangle T(n,k) begins:
   1;
   1,  1;
   2,  2,  2;
   2,  2,  2,  2;
   4,  3,  5,  3,  4;
   4,  4,  4,  4,  4,  4;
   7,  5,  7,  8,  7,  5,  7;
   8,  7,  7,  8,  8,  7,  7,  8;
  12,  9, 12,  9, 17,  9, 12,  9, 12;
  ...
From _Gus Wiseman_, Oct 11 2023: (Start)
Row n = 8 counts the following partitions:
  (8)     (8)    (8)     (8)     (8)     (8)    (8)
  (62)    (71)   (71)    (71)    (71)    (71)   (62)
  (53)    (53)   (62)    (62)    (62)    (53)   (53)
  (44)    (44)   (611)   (611)   (611)   (44)   (44)
  (422)   (431)  (44)    (53)    (44)    (431)  (422)
  (332)          (422)   (521)   (422)          (332)
  (2222)         (2222)  (5111)  (2222)         (2222)
                         (332)
(End)

Alois P. Heinz, Rows n = 2..98, flattened
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Column k = 0 and diagonal k = n are both A002865.
Central diagonal n = 2k is A006827.
The complement with expanded domain is A365543.
The strict case is A365663, complement A365661.
Row sums are A365918, complement A304792.
For subsets instead of partitions we have A366320, complement A365381.
A000041 counts integer partitions, strict A000009.
A276024 counts distinct subset-sums of partitions.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000124, A002219, A093971, A108917, A122768, A299701, A365658.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(0 in s or n in s, 0, `if`(n=0 or s={}, g(n, i),
     `if`(i<1, 0, b(n, i-1, s)+`if`(i>n, 0, b(n-i, i,
      select(y-> 0<=y and y<=n-i, map(x-> [x, x-i][], s)))))))
    end:
T:= (n, k)-> b(n, n, {min(k, n-k)}):
seq(seq(T(n, k), k=1..n-1), n=2..16);  # Alois P. Heinz, Jul 13 2012

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i > 1, g[n, i-1], 0] + If[i > n, 0, g[n-i, i]]]; b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0 || s == {}, g[n, i], If[i < 1, 0, b[n, i-1, s] + If[i > n, 0, b[n-i, i, Select[Flatten[s /. x_ :> {x, x-i}], 0 <= # <= n-i &]]]]]]; t[n_, k_] := b[n, n, {Min[k, n-k]}]; Table[t[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Aug 20 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#],k]&]],{n,2,10},{k,1,n-1}] (* Gus Wiseman, Oct 11 2023 *)

Corrected and extended by Don Reble, Nov 04 2001

A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12

Offset: 0

Gus Wiseman, Sep 16 2023

First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
Rows are palindromic.
Are there only two zeros in the whole triangle?

			Triangle begins:
  1
  1  1
  1  0  1
  2  1  1  2
  2  1  0  1  2
  3  1  1  1  1  3
  4  2  2  1  2  2  4
  5  2  2  2  2  2  2  5
  6  3  2  3  1  3  2  3  6
  8  3  3  4  3  3  4  3  3  8
Row n = 6 counts the following strict partitions:
  (6)      (5,1)    (4,2)    (3,2,1)  (4,2)    (5,1)    (6)
  (5,1)    (3,2,1)  (3,2,1)           (3,2,1)  (3,2,1)  (5,1)
  (4,2)                                                 (4,2)
  (3,2,1)                                               (3,2,1)
Row n = 10 counts the following strict partitions:
  A     91    82    73    64    532   64    73    82    91    A
  64    541   532   532   541   541   541   532   532   541   64
  73    631   721   631   631   4321  631   631   721   631   73
  82    721   4321  721   4321        4321  721   4321  721   82
  91    4321        4321                    4321        4321  91
  532                                                         532
  541                                                         541
  631                                                         631
  721                                                         721
  4321                                                        4321

Robert Price, Table of n, a(n) for n = 0..1325

Columns k = 0 and k = n are A000009.
The non-strict complement is A046663, central column A006827.
Central column n = 2k is A237258.
For subsets instead of partitions we have A365381.
The non-strict case is A365543.
The complement is A365663.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#],k]&]], {n,0,10},{k,0,n}]

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Gus Wiseman, Sep 17 2023

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)

Robert Price, Table of n, a(n) for n = 2..1226
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

A237258 Number of strict partitions of 2n that include a partition of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 32, 43, 63, 84, 122, 158, 220, 293, 393, 511, 685, 881, 1156, 1485, 1925, 2445, 3147, 3952, 5019, 6323, 7924, 9862, 12336, 15259, 18900, 23294, 28646, 35091, 42985, 52341, 63694, 77336, 93588, 112973, 136367, 163874, 196638

Offset: 0

Clark Kimberling, Feb 05 2014

nonn

A strict partition is a partition into distinct parts.

			a(5) counts these partitions of 10: [5,4,1], [5,3,2], [4,3,2,1].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Cf. A000009, A237194, A235130.
The non-strict version is A002219, ranked by A357976.
These partitions are ranked by A357854.
A000712 counts distinct submultisets of partitions, strict A032302.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A006827, A064914, A108917, A276107, A300061, A357879.

z = 24; Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &,  Select[IntegerPartitions[2 nn], # == DeleteDuplicates[#] &]]; Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], nn]] >= 1 &]]], {nn, z}] (* Peter J. C. Moses, Feb 04 2014 *)

a(n) = A237194(2n,n).

a(31)-a(47) from Alois P. Heinz, Feb 07 2014

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Gus Wiseman, Sep 16 2023

Conjecture: Positions of strictly positive rows are given by A048166.

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71

Robert Price, Table of n, a(n) for n = 1..300

Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]

cp's OEIS Frontend

A276024 Number of positive subset sums of integer partitions of n.

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A126796 Number of complete partitions of n.

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A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

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A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

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A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

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A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

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A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

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