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
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)
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.
-
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}]
A336139
Number of ways to choose a strict composition of each part of a strict composition of n.
Original entry on oeis.org
1, 1, 1, 5, 9, 17, 45, 81, 181, 397, 965, 1729, 3673, 7313, 15401, 34065, 68617, 135069, 266701, 556969, 1061921, 2434385, 4436157, 9120869, 17811665, 35651301, 68949549, 136796317, 283612973, 537616261, 1039994921, 2081261717, 3980842425, 7723253181, 15027216049
Offset: 0
The a(1) = 1 through a(5) = 17 splittings:
(1) (2) (3) (4) (5)
(1,2) (1,3) (1,4)
(2,1) (3,1) (2,3)
(1),(2) (1),(3) (3,2)
(2),(1) (3),(1) (4,1)
(1),(1,2) (1),(4)
(1),(2,1) (2),(3)
(1,2),(1) (3),(2)
(2,1),(1) (4),(1)
(1),(1,3)
(1,2),(2)
(1),(3,1)
(1,3),(1)
(2),(1,2)
(2,1),(2)
(2),(2,1)
(3,1),(1)
The version for partitions is
A063834.
The version for non-strict compositions is
A133494.
The version for strict partitions is
A279785.
Multiset partitions of partitions are
A001970.
Taking a composition of each part of a partition:
A075900.
Taking a composition of each part of a strict partition:
A304961.
Taking a strict composition of each part of a composition:
A307068.
Splittings of partitions are
A323583.
Compositions of parts of strict compositions are
A336127.
Set partitions of strict compositions are
A336140.
-
strs[n_]:=Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[strs/@ctn],{ctn,strs[n]}]],{n,0,15}]
A307068
Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)).
Original entry on oeis.org
1, 1, 2, 6, 14, 34, 88, 216, 532, 1322, 3290, 8142, 20192, 50080, 124144, 307878, 763474, 1893038, 4694060, 11639580, 28861736, 71567206, 177460750, 440037738, 1091134276, 2705618900, 6708953156, 16635775698, 41250705518, 102286806130, 253634237896, 628921097352, 1559496588628
Offset: 0
From _Gus Wiseman_, Jul 18 2020: (Start)
The a(1) = 1 through a(4) = 14 ways to choose a strict composition of each part of a composition:
(1) (2) (3) (4)
(1),(1) (1,2) (1,3)
(2,1) (3,1)
(1),(2) (1),(3)
(2),(1) (2),(2)
(1),(1),(1) (3),(1)
(1),(1,2)
(1),(2,1)
(1,2),(1)
(2,1),(1)
(1),(1),(2)
(1),(2),(1)
(2),(1),(1)
(1),(1),(1),(1)
(End)
The version for partitions is
A270995.
Starting with a strict composition gives
A336139.
Strict compositions are counted by
A032020.
Partitions of each part of a partition are
A063834.
Compositions of each part of a partition are
A075900.
Compositions of each part of a composition are
A133494.
Strict partitions of each part of a strict partition are
A279785.
Compositions of each part of a strict partition are
A304961.
Strict partitions of each part of a composition are
A304969.
Compositions of each part of a strict composition are
A336127.
Set partitions of strict compositions are
A336140.
Strict compositions of each part of a partition are
A336141.
-
m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(1 - (&+[Factorial(k)*x^Binomial(k+1,2)/(&*[ 1-x^j: j in [1..k]]): k in [1..m+2]]) ) )); // G. C. Greubel, Jan 25 2024
-
T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
end:
g:= proc(n) option remember; add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)) end:
a:= proc(n) option remember; `if`(n<1, 1,
add(a(n-i)*g(i), i=1..n))
end:
seq(a(n), n=0..32); # Alois P. Heinz, Dec 16 2022
-
nmax = 32; CoefficientList[Series[1/(1 - Sum[k!*x^(k*(k+1)/2)/Product[ (1-x^j), {j,k}], {k,nmax}]), {x, 0, nmax}], x]
-
m=80;
def p(x, j): return product(1-x^k for k in range(1,j+1))
def f(x): return 1/(1 - sum(factorial(j)*x^binomial(j+1,2)/p(x,j) for j in range(1, m+3)) )
def A307068_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( f(x) ).list()
A307068_list(m) # G. C. Greubel, Jan 25 2024
A358907
Number of finite sequences of distinct integer compositions with total sum n.
Original entry on oeis.org
1, 1, 2, 8, 18, 54, 156, 412, 1168, 3200, 8848, 24192, 66632, 181912, 495536, 1354880, 3680352, 9997056, 27093216, 73376512, 198355840, 535319168, 1443042688, 3884515008, 10445579840, 28046885824, 75225974912, 201536064896, 539339293824, 1441781213952
Offset: 0
The a(1) = 1 through a(4) = 18 sequences:
((1)) ((2)) ((3)) ((4))
((11)) ((12)) ((13))
((21)) ((22))
((111)) ((31))
((1)(2)) ((112))
((2)(1)) ((121))
((1)(11)) ((211))
((11)(1)) ((1111))
((1)(3))
((3)(1))
((1)(12))
((11)(2))
((1)(21))
((12)(1))
((2)(11))
((21)(1))
((1)(111))
((111)(1))
This is the strict case of
A133494.
The version for sequences of partitions is
A358906.
A001970 counts multiset partitions of integer partitions.
A218482 counts sequences of compositions with weakly decreasing lengths.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all different Omegas.
A358914 counts twice-partitions into distinct strict partitions.
Cf.
A000009,
A000041,
A000219,
A055887,
A075900,
A296122,
A304961,
A307068,
A336342,
A358836,
A358912.
-
g:= proc(n) option remember; ceil(2^(n-1)) end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32); # Alois P. Heinz, Dec 15 2022
-
comps[n_]:=Join@@Permutations/@IntegerPartitions[n];
Table[Length[Select[Join@@Table[Tuples[comps/@c],{c,comps[n]}],UnsameQ@@#&]],{n,0,10}]
A336343
Number of ways to choose a strict partition of each part of a strict composition of n.
Original entry on oeis.org
1, 1, 1, 4, 6, 11, 26, 39, 78, 142, 320, 488, 913, 1558, 2798, 5865, 9482, 16742, 28474, 50814, 82800, 172540, 266093, 472432, 790824, 1361460, 2251665, 3844412, 7205416, 11370048, 19483502, 32416924, 54367066, 88708832, 149179800, 239738369, 445689392
Offset: 0
The a(1) = 1 through a(5) = 11 ways:
(1) (2) (3) (4) (5)
(2,1) (3,1) (3,2)
(1),(2) (1),(3) (4,1)
(2),(1) (3),(1) (1),(4)
(1),(2,1) (2),(3)
(2,1),(1) (3),(2)
(4),(1)
(1),(3,1)
(2,1),(2)
(2),(2,1)
(3,1),(1)
Multiset partitions of partitions are
A001970.
Splittings of strict partitions are
A072706.
Set partitions of strict partitions are
A294617.
Splittings of partitions with distinct sums are
A336131.
Cf.
A008289,
A011782,
A304786,
A318683,
A318684,
A319794,
A323583,
A336128,
A336130,
A336132,
A336133.
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.
-
strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[strptn/@ctn],{ctn,Join@@Permutations/@strptn[n]}]],{n,0,10}]
-
\\ here Q(N) gives A000009 as a vector.
Q(n) = {Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)))}
seq(n)={my(b=Q(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*b[1+k] + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021
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
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)
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.
-
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 *)
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.
-
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.
-
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 *)
A358904
Number of finite sets of compositions with all equal sums and total sum n.
Original entry on oeis.org
1, 1, 2, 4, 9, 16, 38, 64, 156, 260, 632, 1024, 2601, 4096, 10208, 16944, 40966, 65536, 168672, 262144, 656980, 1090240, 2620928, 4194304, 10862100, 16781584, 41940992, 69872384, 168403448, 268435456, 693528552, 1073741824, 2695006177, 4473400320, 10737385472
Offset: 0
The a(1) = 1 through a(4) = 9 sets:
{(1)} {(2)} {(3)} {(4)}
{(11)} {(12)} {(13)}
{(21)} {(22)}
{(111)} {(31)}
{(112)}
{(121)}
{(211)}
{(1111)}
{(2),(11)}
The case of sets of partitions is
A359041.
A001970 counts multisets of partitions.
-
Table[If[n==0,1,Sum[Binomial[2^(d-1),n/d],{d,Divisors[n]}]],{n,0,30}]
-
a(n) = if (n, sumdiv(n, d, binomial(2^(d-1), n/d)), 1); \\ Michel Marcus, Dec 14 2022
A355387
Number of ways to choose a distinct subsequence of an integer composition of n.
Original entry on oeis.org
1, 2, 5, 14, 37, 98, 259, 682, 1791, 4697, 12303, 32196, 84199, 220087, 575067, 1502176, 3923117, 10244069, 26746171, 69825070, 182276806, 475804961, 1241965456, 3241732629, 8461261457, 22084402087, 57640875725, 150442742575, 392652788250, 1024810764496
Offset: 0
The a(3) = 14 pairings of a composition with a chosen subsequence:
(3)() (3)(3)
(21)() (21)(1) (21)(2) (21)(21)
(12)() (12)(1) (12)(2) (12)(12)
(111)() (111)(1) (111)(11) (111)(111)
The case of strict subsequences is
A236002.
The composable case is
A355384, homogeneous without containment
A355388.
A075900 counts compositions of each part of a partition.
A304961 counts compositions of each part of a strict partition.
A307068 counts strict compositions of each part of a composition.
A336127 counts compositions of each part of a strict composition.
Cf.
A011782,
A022811,
A032020,
A063834,
A133494,
A181591,
A323583,
A331330,
A336128,
A336130,
A336139,
A355382,
A355383.
-
Table[Sum[Length[Union[Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,6}]
-
lista(n)=my(f=sum(k=1,n,(x^k+x*O(x^n))/(1-x/(1-x)+x^k)));Vec((1-x)/((1-2*x)*(1-f))) \\ Christian Sievers, May 06 2025
Comments