A122768 -id:A122768 - cp's OEIS frontend

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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

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

A304792 Number of subset-sums of integer partitions of n.

Original entry on oeis.org

1, 2, 5, 10, 19, 34, 58, 96, 152, 240, 361, 548, 795, 1164, 1647, 2354, 3243, 4534, 6150, 8420, 11240, 15156, 19938, 26514, 34513, 45260, 58298, 75704, 96515, 124064, 157072, 199894, 251097, 317278, 395625, 496184, 615229, 765836, 944045, 1168792, 1432439

Offset: 0

Gus Wiseman, May 18 2018

nonn

For a multiset p of positive integers summing to n, a pair (t,p) is defined to be a subset sum if there exists a submultiset of p summing to t. This sequence is dominated by A122768 + A000041 (number of submultisets of integer partitions of n).

			The a(4)=19 subset sums are (0,4), (4,4), (0,31), (1,31), (3,31), (4,31), (0,22), (2,22), (4,22), (0,211), (1,211), (2,211), (3,211), (4,211), (0,1111), (1,1111), (2,1111), (3,1111), (4,1111).

Robert Price, Table of n, a(n) for n = 0..70

Cf. A000041, A108917, A122768, A276024, A284640, A299701, A301856, A301934, A301979, A304793.

b:= proc(n, i, s) option remember; `if`(n=0, nops(s),
     `if`(i<1, 0, b(n, i-1, s)+b(n-i, min(n-i, i),
      map(x-> [x, x+i][], s))))
    end:
a:= n-> b(n$2, {0}):
seq(a(n), n=0..40);  # Alois P. Heinz, May 18 2018

Table[Total[Length[Union[Total/@Subsets[#]]]&/@IntegerPartitions[n]],{n,15}]
(* 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}];
a /@ Range[0, 40] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A304792_T(n,i,s,l):
    if n==0: return l
    if i<1: return 0
    return A304792_T(n,i-1,s,l)+A304792_T(n-i,min(n-i,i),(t:=tuple(sorted(set(s+tuple(x+i for x in s))))),len(t))
def A304792(n): return A304792_T(n,n,(0,),1) # Chai Wah Wu, Sep 25 2023, after Alois P. Heinz

a(n) = A276024(n) + A000041(n).

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

A284640 Number of positive subset sums of strict integer partitions of n.

Original entry on oeis.org

1, 1, 4, 4, 7, 13, 17, 23, 34, 49, 62, 87, 112, 149, 199, 249, 318, 408, 512, 635, 820, 991, 1238, 1515, 1864, 2248, 2770, 3326, 4030, 4818, 5808, 6882, 8290, 9756, 11639, 13719, 16236, 18999, 22468, 26144, 30724, 35761, 41754, 48357, 56380, 65018, 75438

Offset: 1

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Author

Gus Wiseman, Mar 31 2017

Keywords

nonn

Comments

For a strict integer partition p summing to n, a pair (t,p) is defined to be a positive subset sum if there exists a nonempty subset of p summing to t.

Examples

The a(6)=13 subset sums are: (6,6), (1,51), (5,51), (6,51), (2,42), (4,42), (6,42), (1,321), (2,321), (3,321), (4,321), (5,321), (6,321).

Links

Robert Price, Table of n, a(n) for n = 1..80

Crossrefs

Cf. A122768, A275972, A276024.

Programs

Mathematica

nn=25;Total/@Table[Function[ptn,Length[Union[Total/@Rest[Subsets[ptn]]]]]/@Select[IntegerPartitions[n],UnsameQ@@#&],{n,nn}]

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

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Author

Gus Wiseman, Sep 16 2023

Keywords

nonn
tabl

Comments

Rows are palindromic.

Examples

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)

Links

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

Crossrefs

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.

Programs

Mathematica

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

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Author

N. J. A. Sloane

Keywords

nonn
easy
look
nice
tabl

Examples

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)

Links

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.

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

Extensions

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

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Author

Gus Wiseman, Sep 16 2023

Keywords

nonn
tabl

Comments

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?

Examples

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

Links

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

Crossrefs

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.

Programs

Mathematica

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

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Author

Gus Wiseman, Sep 17 2023

Keywords

nonn
tabl

Comments

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

Rows are palindromes.

Examples

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)

Links

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.

Crossrefs

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.

Programs

Mathematica

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

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

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Internal format

Author

Alois P. Heinz, Sep 09 2008

Keywords

nonn
tabl

Comments

A(n,k) is also the number of partitions of n into parts of k kinds.

In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

Examples

Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, ... 0, 2, 5, 9, 14, 20, ... 0, 3, 10, 22, 40, 65, ... 0, 5, 20, 51, 105, 190, ... 0, 7, 36, 108, 252, 506, ...

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)

G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

N. J. A. Sloane, Transforms

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.

Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).

Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.

Main diagonal gives A008485.

Antidiagonal sums give A067687.

Cf. A060642, A122768, A246935, A255961, A261718.

Programs

Julia

# DedekindEta is defined in A000594. A144064Column(k, len) = DedekindEta(len, -k) for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

Maple

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *) etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]} ]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

PARI

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

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A276024 Number of positive subset sums of integer 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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A304792 Number of subset-sums of integer partitions of 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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A284640 Number of positive subset sums of strict integer 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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