A316245 -id:A316245 - cp's OEIS frontend

A336135 Number of ways to split an integer partition of n into contiguous subsequences with strictly decreasing sums.

1, 1, 2, 5, 8, 16, 29, 50, 79, 135, 213, 337, 522, 796, 1191, 1791, 2603, 3799, 5506, 7873, 11154, 15768, 21986, 30565, 42218, 57917, 78968, 107399, 144932, 194889, 261061, 347773, 461249, 610059, 802778, 1053173, 1377325, 1793985, 2329009, 3015922, 3891142

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(5) = 16 splittings:
  (1)  (2)    (3)        (4)          (5)
       (1,1)  (2,1)      (2,2)        (3,2)
              (1,1,1)    (3,1)        (4,1)
              (2),(1)    (2,1,1)      (2,2,1)
              (1,1),(1)  (3),(1)      (3,1,1)
                         (1,1,1,1)    (3),(2)
                         (2,1),(1)    (4),(1)
                         (1,1,1),(1)  (2,1,1,1)
                                      (2,2),(1)
                                      (3),(1,1)
                                      (3,1),(1)
                                      (1,1,1,1,1)
                                      (2,1),(1,1)
                                      (2,1,1),(1)
                                      (1,1,1),(1,1)
                                      (1,1,1,1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..50

The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with weakly increasing sums is A336136.
The version with weakly decreasing sums is A316245.
The version with different sums is A336131.
Starting with a composition gives A304961.
Starting with a strict partition gives A318684.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A323433, A336128, A336130, A336133.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Greater@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f, self()(r,min(m,t-1),t-1,0,0)) + self()(r,m-1,s,t,0) + if(t+m<=s, self()(r-m,min(m,r-m),s,t+m,1)))); recurse(n,n,n,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A336342 Number of ways to choose a partition of each part of a strict composition of n.

Original entry on oeis.org

1, 1, 2, 7, 11, 29, 81, 155, 312, 708, 1950, 3384, 7729, 14929, 32407, 81708, 151429, 305899, 623713, 1234736, 2463743, 6208978, 10732222, 22487671, 43000345, 86573952, 160595426, 324990308, 744946690, 1336552491, 2629260284, 5050032692, 9681365777

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

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

Is there a simple generating function?

			The a(1) = 1 through a(4) = 11 ways:
  (1)  (2)    (3)        (4)
       (1,1)  (2,1)      (2,2)
              (1,1,1)    (3,1)
              (1),(2)    (1),(3)
              (2),(1)    (2,1,1)
              (1),(1,1)  (3),(1)
              (1,1),(1)  (1,1,1,1)
                         (1),(2,1)
                         (2,1),(1)
                         (1),(1,1,1)
                         (1,1,1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A316245, A317715, A336128, A336132, A336134, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

Table[Length[Join@@Table[Tuples[IntegerPartitions/@ctn],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,10}]

seq(n)={[subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*numbpart(k) + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000041(j)). - Andrew Howroyd, Apr 16 2021

A336132 Number of ways to split a strict integer partition of n into contiguous subsequences all having different sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 11, 14, 21, 30, 37, 51, 66, 86, 120, 146, 186, 243, 303, 378, 495, 601, 752, 927, 1150, 1395, 1741, 2114, 2571, 3134, 3788, 4541, 5527, 6583, 7917, 9511, 11319, 13448, 16040, 18996, 22455, 26589, 31317, 36844, 43518, 50917, 59655, 69933

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(7) = 14 splits:
  (1)  (2)  (3)      (4)      (5)      (6)          (7)
            (2,1)    (3,1)    (3,2)    (4,2)        (4,3)
            (2),(1)  (3),(1)  (4,1)    (5,1)        (5,2)
                              (3),(2)  (3,2,1)      (6,1)
                              (4),(1)  (4),(2)      (4,2,1)
                                       (5),(1)      (4),(3)
                                       (3,2),(1)    (5),(2)
                                       (3),(2),(1)  (6),(1)
                                                    (4),(2,1)
                                                    (4,2),(1)
                                                    (4),(2),(1)

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

The version with equal instead of different sums is A318683.
Starting with a composition gives A336127.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336131 Number of ways to split an integer partition of n into contiguous subsequences all having different sums.

Original entry on oeis.org

1, 1, 2, 6, 9, 20, 44, 74, 123, 231, 441, 681, 1188, 1889, 3110, 5448, 8310, 13046

Offset: 0

Gus Wiseman, Jul 11 2020

			The a(1) = 1 through a(4) = 9 splits:
  (1)  (2)    (3)        (4)
       (1,1)  (2,1)      (2,2)
              (1,1,1)    (3,1)
              (2),(1)    (2,1,1)
              (1),(1,1)  (3),(1)
              (1,1),(1)  (1,1,1,1)
                         (2,1),(1)
                         (1),(1,1,1)
                         (1,1,1),(1)

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

The version with equal instead of different sums is A317715.
Starting with a composition gives A336127.
Starting with a strict composition gives A336128.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 27, 37, 62, 82, 125, 168, 246, 320, 462, 585, 839, 1078, 1466, 1830, 2528, 3136, 4188, 5210, 6907, 8498, 11177, 13570, 17668, 21614, 27580, 33339, 42817, 51469, 65083, 78457, 98409, 117602, 147106, 174663, 217400, 259318, 319076, 377707

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(6) = 17 splits:
  (1)  (2)    (3)        (4)          (5)            (6)
       (1,1)  (2,1)      (2,2)        (3,2)          (3,3)
              (1,1,1)    (3,1)        (4,1)          (4,2)
              (1),(1,1)  (2,1,1)      (2,2,1)        (5,1)
                         (1,1,1,1)    (3,1,1)        (2,2,2)
                         (1),(1,1,1)  (2,1,1,1)      (3,2,1)
                                      (2),(2,1)      (4,1,1)
                                      (1,1,1,1,1)    (2,2,1,1)
                                      (2),(1,1,1)    (2),(2,2)
                                      (1),(1,1,1,1)  (3,1,1,1)
                                      (1,1),(1,1,1)  (2,1,1,1,1)
                                                     (2),(2,1,1)
                                                     (1,1,1,1,1,1)
                                                     (2),(1,1,1,1)
                                                     (1),(1,1,1,1,1)
                                                     (1,1),(1,1,1,1)
                                                     (1),(1,1),(1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..75

The version with equal sums is A317715.
The version with strictly decreasing sums is A336135.
The version with weakly decreasing sums is A316245.
The version with different sums is A336131.
Starting with a composition gives A304961.
Starting with a strict partition gives A336133.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A318684, A323433, A336128, A336130.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r > t && t >= s, self()(r,m,t+1,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m), s,t+m,1))); recurse(n,n,0,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A318434 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(3072) = 5 constant-sum split partitions:
  (21111111111)
  (21111)(111111)
  (211)(1111)(1111)
  (21)(111)(111)(111)
  (2)(11)(11)(11)(11)(11)

Cf. A001970, A056239, A063834, A296150, A316223, A317545, A317546, A319002.

Cf. A316245, A317508, A317715, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Length[Select[comps[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],SameQ@@Total/@#&]],{n,100}]

A317508 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(60) = 7 split partitions:
  (3)(2)(1)(1)
  (32)(1)(1)
  (3)(21)(1)
  (3)(2)(11)
  (321)(1)
  (32)(11)
  (3211)

Cf. A001970, A056239, A063834, A255397, A296150, A316223, A317545, A317546, A319002, A319004.

Cf. A316245, A317715, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Length[Select[compositionPartitions[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],OrderedQ[Total/@#]&]],{n,100}]

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 16, 26, 42, 64, 100, 150, 224, 330, 482, 697, 999, 1418, 1996, 2794, 3879, 5355, 7343, 10018, 13583, 18338, 24618, 32917, 43790, 58043, 76591, 100716, 131906, 172194, 223966, 290423, 375318, 483668, 621368, 796138, 1017146

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(1)  (21)(21)  (31)(31)   (41)(41)
                         (111)(1)  (22)(2)    (32)(32)
                                   (211)(11)  (311)(11)
                                   (211)(21)  (311)(31)
                                   (1111)(1)  (221)(21)
                                              (221)(22)
                                              (2111)(11)
                                              (2111)(21)
                                              (11111)(1)

With multiplicity we have A339006.
The version for compositions is A355384.
The homogeneous version w/o containment is A355385, compositions A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.
Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]

A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 20, 24, 31, 38, 48, 59, 72, 86, 106, 125, 150, 180, 213, 250, 296, 347, 407, 477, 555, 645, 751, 869, 1003, 1161, 1334, 1534, 1763, 2018, 2306, 2637, 3002, 3418, 3886, 4409, 4994, 5659, 6390, 7214, 8135, 9160, 10300, 11580, 12990

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(10) = 20 split partitions:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 3]
  [1] [2] [3] [4] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Cf. A000005, A000219, A101509, A117433, A316245, A319066, A319794, A323295, A323301, A323431, A323433.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, numtheory[tau](t), b(n, i-1, t)+
         b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
     If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
     b[n-i, Min[n-i, i-1], t+1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.

A336141 Number of ways to choose a strict composition of each part of an integer partition of n.

Original entry on oeis.org

1, 1, 2, 5, 9, 17, 41, 71, 138, 270, 518, 938, 1863, 3323, 6163, 11436, 20883, 37413, 69257, 122784, 221873, 397258, 708142, 1249955, 2236499, 3917628, 6909676, 12130972, 21251742, 36973609, 64788378, 112103360, 194628113, 336713377, 581527210, 1000153063

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

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

			The a(1) = 1 through a(5) = 17 ways:
  (1)  (2)      (3)          (4)              (5)
       (1),(1)  (1,2)        (1,3)            (1,4)
                (2,1)        (3,1)            (2,3)
                (2),(1)      (2),(2)          (3,2)
                (1),(1),(1)  (3),(1)          (4,1)
                             (1,2),(1)        (3),(2)
                             (2,1),(1)        (4),(1)
                             (2),(1),(1)      (1,2),(2)
                             (1),(1),(1),(1)  (1,3),(1)
                                              (2,1),(2)
                                              (3,1),(1)
                                              (2),(2),(1)
                                              (3),(1),(1)
                                              (1,2),(1),(1)
                                              (2,1),(1),(1)
                                              (2),(1),(1),(1)
                                              (1),(1),(1),(1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..5623

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

Table[Length[Join@@Table[Tuples[Join@@Permutations/@Select[IntegerPartitions[#],UnsameQ@@#&]&/@ctn],{ctn,IntegerPartitions[n]}]],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1] +
     b[i, i, 0] g[n-i, Min[n-i, i]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} 1/(1 - A032020(k)*x^k).

cp's OEIS Frontend

A336135 Number of ways to split an integer partition of n into contiguous subsequences with strictly decreasing sums.

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A336342 Number of ways to choose a partition of each part of a strict composition of n.

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A336132 Number of ways to split a strict integer partition of n into contiguous subsequences all having different sums.

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A336131 Number of ways to split an integer partition of n into contiguous subsequences all having different sums.

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A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.

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A318434 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.

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A317508 Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.

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A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

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