A035470 -id:A035470 - cp's OEIS frontend

A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 5, 3, 0, 1, 0, 7, 0, 0, 0, 1, 0, 11, 6, 4, 0, 0, 1, 0, 15, 0, 0, 0, 0, 0, 1, 0, 22, 14, 0, 5, 0, 0, 0, 1, 0, 30, 0, 10, 0, 0, 0, 0, 0, 1, 0, 42, 25, 0, 0, 6, 0, 0, 0, 0, 1, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 77, 53, 30, 15, 0, 7, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 20 2024

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.

			Triangle begins:
  1
  0  1
  0  2  1
  0  3  0  1
  0  5  3  0  1
  0  7  0  0  0  1
  0 11  6  4  0  0  1
  0 15  0  0  0  0  0  1
  0 22 14  0  5  0  0  0  1
  0 30  0 10  0  0  0  0  0  1
  0 42 25  0  0  6  0  0  0  0  1
  0 56  0  0  0  0  0  0  0  0  0  1
  0 77 53 30 15  0  7  0  0  0  0  0  1
Row n = 6 counts the following partitions:
  .  (6)       (33)      (222)     .  .  (111111)
     (51)      (321)     (2211)
     (42)      (3111)    (21111)
     (411)     (2211)    (111111)
     (33)      (21111)
     (321)     (111111)
     (3111)
     (222)
     (2211)
     (21111)
     (111111)

Row n has A000005(n) positive entries.
Column k = 1 is A000041.
Column k = 2 is A002219 (aerated), ranks A357976.
Column k = 3 is A002220 (aerated), ranks A371955.
Removing all zeros gives A371783.
Row sums are A372121.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, complement A371796.
Cf. A002221, A002222, A006827, A035470, A064914, A237258, A321142, A321455, A365543, A371795.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,0,10},{k,0,n}]

A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

Original entry on oeis.org

1, 3, 4, 9, 8, 22, 16, 42, 41, 74, 57, 183, 102, 233, 263, 463, 298, 875, 491, 1350, 1172, 1775, 1256, 4273, 2225, 4399, 4584, 8049, 4566, 14913, 6843, 18539, 15831, 22894, 18196, 53323, 21638, 48947, 50281, 94500, 44584, 144976, 63262, 173436, 169361, 202153

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums. The triangles A371783 and A371954 count k-quanimous partitions.

Row sums of A371783.
Row sums of A371954.
A000005 counts divisors.
A000041 counts integer partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A321452 counts quanimous partitions, complement A321451.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A035470, A064914, A321455, A365543, A371737, A371791, A371795.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Sum[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{k,Divisors[n]}],{n,1,10}]

T(n, d) = my(v=partitions(n/d), w=List([])); forvec(s=vector(d, i, [1, #v]), listput(w, vecsort(concat(vector(d, i, v[s[i]])))), 1); #Set(w);
a(n) = sumdiv(n, d, T(n, d)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A164988 Number of ways to select disjoint subsets out of {1..n} such that their (sorted) element sums give the list of divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 7, 2, 10, 9, 7, 9, 21, 8, 29, 12, 31, 67, 56, 11, 79, 167, 105, 85, 137, 37, 181, 248, 346, 893, 299, 106, 404, 1974, 993, 338, 669, 724, 853, 3335, 1068, 8757, 1371, 852, 2422, 9157, 7124, 17168, 2702, 11606, 6390, 10782, 17681, 68538

Offset: 1

Alois P. Heinz, Sep 03 2009

			a(9) = 3: subset selections are [{1},{3},{9}], [{1},{3},{2,7}], [{1},{3},{4,5}].
a(10) = 3: [{1},{2},{5},{10}], [{1},{2},{5},{3,7}], [{1},{2},{5},{4,6}].
a(11) = 7: [{1},{11}], [{1},{2,9}], [{1},{3,8}], [{1},{4,7}], [{1},{5,6}], [{1},{2,3,6}], [{1},{2,4,5}].
a(12) = 2: [{1},{2},{3},{4},{6},{12}], [{1},{2},{3},{4},{6},{5,7}].

Alois P. Heinz, Table of n, a(n) for n = 1..155

Cf. A065205, A035470.

with(numtheory): b:= proc() option remember; local i, j, t, m; m:= args[nargs]; if nargs=1 then 1 elif args[1]=0 then b(args[t] $t=2..nargs) elif m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2 then 0 else b(args[t] $t=1..nargs-1, m-1) +add(`if`(args[j]-m<0, 0, b(sort([seq(args[i] -`if`(i=j, m, 0), i=1..nargs-1)])[], m-1)), j=1..nargs-1) fi end: a:= n-> b(divisors(n)[], n): seq(a(n), n=1..40);

$RecursionLimit = 1000; b[args__] := b[args] = Module[{i, j, t, m, nargs}, nargs = Length[{args}]; m = Last[{args}]; Which [nargs == 1, 1, {args}[[1]] == 0, b @@ Rest[{args}], m == 0 || Total[Most[{args}]] > m*(m+1)/2, 0, True, b[Sequence @@ Most[{args}], m-1] + Sum [If[{args}[[j]] - m < 0, 0, b[Sequence @@ Sort[Table[{args}[[i]] - If [i == j, m, 0], {i, 1, nargs-1}]], m-1]], {j, 1, nargs-1}]] ]; a[n_] := b[Sequence @@ Divisors[n], n]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 95}] (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

a(p) = A025147(p) for p prime. - Charlie Neder, Jan 15 2019

A320438 Irregular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with all block-sums equal to d, where d is the k-th divisor of n*(n+1)/2 that is >= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 3, 7, 1, 1, 9, 1, 1, 1, 1, 43, 35, 1, 1, 102, 62, 1, 1, 1, 1, 68, 595, 1, 1, 17, 187, 871, 1480, 361, 1, 1, 2650, 657, 1, 1, 9294, 1, 1, 23728, 1, 1, 27763, 4110, 1, 1, 1850, 25035, 108516, 157991, 7636, 1, 1, 11421, 411474, 1

Offset: 1

Gus Wiseman, Jan 08 2019

			Triangle begins:
    1
    1
    1    1
    1    1
    1    1
    1    1
    1    4    1
    1    3    7    1
    1    9    1
    1    1
    1   43   35    1
    1  102   62    1
    1    1
    1   68  595    1
    1   17  187  871 1480  361    1
    1 2650  657    1
Row 8 counts the following set partitions:
  {{18}{27}{36}{45}}  {{1236}{48}{57}}  {{12348}{567}}  {{12345678}}
                      {{138}{246}{57}}  {{12357}{468}}
                      {{156}{237}{48}}  {{12456}{378}}
                                        {{1278}{3456}}
                                        {{1368}{2457}}
                                        {{1458}{2367}}
                                        {{1467}{2358}}

Row lengths are A164978. Row sums are A035470.

Cf. A000110, A000258, A008277, A112956, A164977, A275714, A279375, A300335, A320423, A320424, A321455, A321469.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Total[#]==d&],Range[n]]],{n,12},{d,Select[Divisors[n*(n+1)/2],#>=n&]}]

More terms from Jinyuan Wang, Feb 27 2025

Name edited by Peter Munn, Mar 06 2025

A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

Original entry on oeis.org

1, 2, 3, 9, 26, 69, 335, 1018, 6629, 22805, 182988, 703745

Offset: 1

Gus Wiseman, Apr 04 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Since (n+1)/2 is the median of {1..n}, this sequence counts "transitive" set partitions.

			The a(1) = 1 through a(4) = 9 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{23}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with median 5/2, so is counted under a(4).
The set partition {{1,3},{2,4}} has medians {2,3}, with median 5/2, so is counted under a(4).

For mean instead of median we have A361910.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A325347 counts partitions w/ integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361864 counts set partitions with integer median of medians, means A361865.
A361866 counts set partitions with integer sum of medians, means A361911.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A326516, A326521, A326537, A359907, A361654, A361801.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],(n+1)/2==Median[Median/@#]&]],{n,6}]

A372122 Number of strict triquanimous partitions of 3n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 5, 13, 18, 36, 51, 93, 132, 229, 315, 516, 735, 1134, 1575, 2407, 3309, 4878, 6710, 9690, 13168, 18744, 25114, 35050, 47210, 64503, 85573, 116445, 153328, 205367, 269383, 356668, 464268, 610644, 788274, 1026330, 1321017, 1704309, 2176054

Offset: 0

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums. Triquanimous partitions are counted by A002220 and ranked by A371955.

			The partition (11,7,5,4,3,2,1) has qualifying set partitions {{11},{4,7},{1,2,3,5}} and {{11},{1,3,7},{2,4,5}} so is counted under a(11).
The a(5) = 1 through a(9) = 13 partitions:
  (5,4,3,2,1)  (6,5,4,2,1)  (7,5,4,3,2)    (8,6,5,3,2)    (9,6,5,4,3)
                            (7,6,4,3,1)    (8,7,5,3,1)    (9,7,5,4,2)
                            (7,6,5,2,1)    (8,7,6,2,1)    (9,7,6,3,2)
                            (6,5,4,3,2,1)  (7,6,5,3,2,1)  (9,8,5,4,1)
                                           (8,6,4,3,2,1)  (9,8,6,3,1)
                                                          (9,8,7,2,1)
                                                          (7,6,5,4,3,2)
                                                          (8,6,5,4,3,1)
                                                          (8,7,5,4,2,1)
                                                          (8,7,6,3,2,1)
                                                          (9,6,5,4,2,1)
                                                          (9,7,5,3,2,1)
                                                          (9,8,4,3,2,1)

The non-strict biquanimous version is A002219, ranks A357976.
The non-strict version is A002220, ranks A371955.
The biquanimous version is A237258, ranks A357854.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454, strict A371737.
A371783 counts k-quanimous partitions.
A371795 counts non-biquanimous partitions, even case A006827, ranks A371731.
Cf. A035470, A064914, A321142, A321455, A371781, A371796.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[3n], UnsameQ@@#&&Select[facs[Times@@Prime/@#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 30 2025

A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 288, 512, 576, 1024, 2048, 3072, 4096, 8192, 16384, 32768, 32800, 33024, 33056, 65536, 65600, 66048, 66112, 131072, 132096, 133120, 134144, 262144, 266240, 524288, 528384, 786432, 790528, 1048576, 1056768, 2097152

Offset: 1

Gus Wiseman, Apr 13 2024

			The terms together with their binary expansions and binary indices begin:
        1:                1 ~ {1}
        2:               10 ~ {2}
        4:              100 ~ {3}
        8:             1000 ~ {4}
       12:             1100 ~ {3,4}
       16:            10000 ~ {5}
       32:           100000 ~ {6}
       64:          1000000 ~ {7}
      128:         10000000 ~ {8}
      144:         10010000 ~ {5,8}
      256:        100000000 ~ {9}
      288:        100100000 ~ {6,9}
      512:       1000000000 ~ {10}
      576:       1001000000 ~ {7,10}
     1024:      10000000000 ~ {11}
     2048:     100000000000 ~ {12}
     3072:     110000000000 ~ {11,12}
     4096:    1000000000000 ~ {13}
     8192:   10000000000000 ~ {14}
    16384:  100000000000000 ~ {15}
    32768: 1000000000000000 ~ {16}
    32800: 1000000000100000 ~ {6,16}

For prime instead of binary indices we have A326534.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A321142 and A371794 count non-biquanimous strict partitions.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A357976 ranks the biquanimous partitions counted by A002219 aerated.
A371731 ranks the non-biquanimous partitions counted by A371795, A006827.
Cf. A035470, A038041, A237258, A320324, A321453, A321455, A326518, A336137, A371783, A371791, A371796.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]

cp's OEIS Frontend

A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

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A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

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A164988 Number of ways to select disjoint subsets out of {1..n} such that their (sorted) element sums give the list of divisors of n.

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A320438 Irregular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with all block-sums equal to d, where d is the k-th divisor of n*(n+1)/2 that is >= n.

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A361863 Number of set partitions of {1..n} such that the median of medians of the blocks is (n+1)/2.

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A372122 Number of strict triquanimous partitions of 3n.

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A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

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