A336133
Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720
Offset: 0
The a(1) = 1 through a(9) = 9 splittings:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4)
(4,1) (5,1) (5,2) (6,2) (6,3)
(3,2,1) (6,1) (7,1) (7,2)
(4,2,1) (4,3,1) (8,1)
(5,2,1) (4,3,2)
(5,3,1)
(6,2,1)
(4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.
The version with equal sums is
A318683.
The version with strictly decreasing sums is
A318684.
The version with weakly decreasing sums is
A319794.
The version with different sums is
A336132.
Starting with a composition gives
A304961.
Starting with a non-strict partition gives
A336134.
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,
A336127,
A336128,
A336130,
A336135.
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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,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]
A336142
Number of ways to choose a strict composition of each part of a strict integer partition of n.
Original entry on oeis.org
1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966
Offset: 0
The a(1) = 1 through a(5) = 11 ways:
(1) (2) (3) (4) (5)
(1,2) (1,3) (1,4)
(2,1) (3,1) (2,3)
(2),(1) (3),(1) (3,2)
(1,2),(1) (4,1)
(2,1),(1) (3),(2)
(4),(1)
(1,2),(2)
(1,3),(1)
(2,1),(2)
(3,1),(1)
Multiset partitions of partitions are
A001970.
Splittings of partitions are
A323583.
Splittings of partitions with distinct sums are
A336131.
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.
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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
-
strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* 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[i(i+1)/2 < n, 0,
If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
A336129
Number of strict compositions of divisors of n.
Original entry on oeis.org
1, 2, 4, 5, 6, 16, 14, 24, 31, 64, 66, 120, 134, 208, 360, 459, 618, 894, 1178, 1622, 2768, 3364, 4758, 6432, 8767, 11440, 15634, 24526, 30462, 42296, 55742, 75334, 98112, 131428, 168444, 258403, 315974, 432244, 558464, 753132, 958266, 1280840, 1621274
Offset: 1
The a(1) = 1 through a(7) = 14 compositions:
(1) (1) (1) (1) (1) (1) (1)
(2) (3) (2) (5) (2) (7)
(1,2) (4) (1,4) (3) (1,6)
(2,1) (1,3) (2,3) (6) (2,5)
(3,1) (3,2) (1,2) (3,4)
(4,1) (1,5) (4,3)
(2,1) (5,2)
(2,4) (6,1)
(4,2) (1,2,4)
(5,1) (1,4,2)
(1,2,3) (2,1,4)
(1,3,2) (2,4,1)
(2,1,3) (4,1,2)
(2,3,1) (4,2,1)
(3,1,2)
(3,2,1)
Compositions of divisors are
A034729.
Strict partitions of divisors are
A047966.
Partitions of divisors are
A047968.
-
Table[Sum[Length[Join@@Permutations/@Select[IntegerPartitions[d],UnsameQ@@#&]],{d,Divisors[n]}],{n,12}]
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