A270995
Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).
Original entry on oeis.org
1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909
Offset: 0
a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.
For compositions instead of partitions we have
A304969, non-strict
A055887.
A072233 counts partitions by sum and length.
-
nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
A055887
Number of ordered partitions of partitions.
Original entry on oeis.org
1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172
Offset: 0
The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
() ((1)) ((2)) ((3))
((11)) ((21))
((1)(1)) ((111))
((1)(2))
((2)(1))
((1)(11))
((11)(1))
((1)(1)(1))
(End)
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
-
with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007
-
a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)
-
Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012
A107742
G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).
Original entry on oeis.org
1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954
Offset: 0
A072233 counts partitions by sum and length.
-
nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* Gus Wiseman, Sep 13 2022 *)
-
a(n)=polcoeff(prod(k=1,n,prod(j=1,n\k,1+x^(j*k)+x*O(x^n))),n) /* Paul D. Hanna */
-
N=66; x='x+O('x^N); gf=1/prod(j=0,N, eta(x^(2*j+1))); gf=prod(j=1,N,(1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */
-
{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */
A358914
Number of twice-partitions of n into distinct strict partitions.
Original entry on oeis.org
1, 1, 1, 3, 4, 7, 13, 20, 32, 51, 83, 130, 206, 320, 496, 759, 1171, 1786, 2714, 4104, 6193, 9286, 13920, 20737, 30865, 45721, 67632, 99683, 146604, 214865, 314782, 459136, 668867, 972425, 1410458, 2040894, 2950839, 4253713, 6123836, 8801349, 12627079
Offset: 0
The a(1) = 1 through a(6) = 13 twice-partitions:
((1)) ((2)) ((3)) ((4)) ((5)) ((6))
((21)) ((31)) ((32)) ((42))
((2)(1)) ((3)(1)) ((41)) ((51))
((21)(1)) ((3)(2)) ((321))
((4)(1)) ((4)(2))
((21)(2)) ((5)(1))
((31)(1)) ((21)(3))
((31)(2))
((3)(21))
((32)(1))
((41)(1))
((3)(2)(1))
((21)(2)(1))
This is the distinct case of
A270995.
The case of strictly decreasing sums is
A279785.
The case of constant sums is
A279791.
For distinct instead of weakly decreasing sums we have
A336343.
This is the twice-partition case of
A358913.
A001970 counts multiset partitions of integer partitions.
A055887 counts sequences of partitions.
A330462 counts set systems by total sum and length.
A358830 counts twice-partitions with distinct lengths.
Cf.
A000009,
A000219,
A075900,
A271619,
A296122,
A304969,
A321449,
A336342,
A358901,
A358906,
A358907.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]
-
seq(n,k)={my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))-1)); Vec(prod(k=1, n, my(c=u[k]); sum(j=0, min(c,n\k), x^(j*k)*c!/(c-j)!, O(x*x^n))))} \\ Andrew Howroyd, Dec 31 2022
A374686
Number of integer compositions of n whose leaders of strictly increasing runs are identical.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 29, 51, 91, 162, 291, 523, 948, 1712, 3112, 5656, 10297, 18763, 34217, 62442, 114006, 208239, 380465, 695342, 1271046, 2323818, 4249113, 7770389, 14210991, 25991853, 47541734, 86962675, 159077005, 291001483, 532345978, 973871397
Offset: 0
The composition (2,3,2,2,3,4) has strictly increasing runs ((2,3),(2),(2,3,4)), with leaders (2,2,2), so is counted under a(16).
The a(0) = 1 through a(6) = 17 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (113) (33)
(121) (131) (114)
(1111) (1112) (123)
(1121) (141)
(1211) (222)
(11111) (1113)
(1131)
(1212)
(1311)
(11112)
(11121)
(11211)
(12111)
(111111)
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have
A000005 for n > 0, ranks
A272919.
- For leaders of weakly increasing runs we have
A374631, ranks
A374633.
- For leaders of weakly decreasing runs we have
A374742, ranks
A374744.
- For leaders of strictly decreasing runs we have
A374760, ranks
A374759.
Types of run-leaders (instead of identical):
- For strictly increasing leaders we have
A374688.
- For strictly decreasing leaders we have
A374689.
- For weakly increasing leaders we have
A374690.
- For weakly decreasing leaders we have
A374697.
A335456 counts patterns matched by compositions.
A374683 lists leaders of strictly increasing runs of standard compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf.
A000009,
A106356,
A188920,
A189076,
A238343,
A304969,
A333213,
A374632,
A374634,
A374635,
A374640.
-
Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@First/@Split[#,Less]&]],{n,0,15}]
-
seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 27 2024
A308680
Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1
Offset: 0
T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 2, 5, 3, 1;
0, 3, 8, 9, 4, 1;
0, 4, 14, 19, 14, 5, 1;
0, 5, 22, 39, 36, 20, 6, 1;
0, 6, 34, 72, 85, 60, 27, 7, 1;
0, 8, 50, 128, 180, 160, 92, 35, 8, 1;
0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
...
Columns k=0-10 give:
A000007,
A000009 (for n>0),
A327380,
A327381,
A327382,
A327383,
A327384,
A327385,
A327386,
A327387,
A327388.
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
T:= proc(n, k) option remember;
`if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
(q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)
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
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)
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.
-
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
A374689
Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.
Original entry on oeis.org
1, 1, 1, 3, 3, 6, 10, 13, 21, 32, 48, 66, 101, 144, 207, 298, 415, 592, 833, 1163, 1615, 2247, 3088, 4259, 5845, 7977, 10862, 14752, 19969, 26941, 36310, 48725, 65279, 87228, 116274, 154660, 205305, 271879, 359400, 474157, 624257, 820450, 1076357, 1409598
Offset: 0
The a(0) = 1 through a(8) = 21 compositions:
() (1) (2) (3) (4) (5) (6) (7) (8)
(12) (13) (14) (15) (16) (17)
(21) (31) (23) (24) (25) (26)
(32) (42) (34) (35)
(41) (51) (43) (53)
(212) (123) (52) (62)
(213) (61) (71)
(231) (124) (125)
(312) (214) (134)
(321) (241) (215)
(313) (251)
(412) (314)
(421) (323)
(341)
(413)
(431)
(512)
(521)
(2123)
(2312)
(3212)
The weak version appears to be
A189076.
Ranked by positions of strictly decreasing rows in
A374683.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have
A000041.
- For leaders of anti-runs we have
A374680.
- For leaders of weakly increasing runs we have
A188920.
- For leaders of weakly decreasing runs we have
A374746.
- For leaders of strictly decreasing runs we have
A374763.
Types of run-leaders (instead of strictly decreasing):
- For strictly increasing leaders we have
A374688.
- For weakly increasing leaders we have
A374690.
- For weakly decreasing leaders we have
A374697.
A335456 counts patterns matched by compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
-
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,Less]&]],{n,0,15}]
-
C_x(N) = {my(x='x+O('x^N), h=prod(i=1,N, 1+(x^i)*prod(j=i+1,N, 1+x^j))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, Jul 29 2024
A356932
Number of multiset partitions of integer partitions of n such that all blocks have odd size.
Original entry on oeis.org
1, 1, 2, 4, 7, 13, 24, 42, 74, 130, 224, 383, 653, 1100, 1846, 3079, 5104, 8418, 13827, 22592, 36774, 59613, 96271, 154908, 248441, 397110, 632823, 1005445, 1592962, 2516905, 3966474, 6235107, 9777791, 15297678, 23880160, 37196958, 57819018, 89691934, 138862937
Offset: 0
The a(1) = 1 through a(5) = 13 multiset partitions:
{1} {2} {3} {4} {5}
{1}{1} {111} {112} {113}
{1}{2} {1}{3} {122}
{1}{1}{1} {2}{2} {1}{4}
{1}{111} {2}{3}
{1}{1}{2} {11111}
{1}{1}{1}{1} {1}{112}
{2}{111}
{1}{1}{3}
{1}{2}{2}
{1}{1}{111}
{1}{1}{1}{2}
{1}{1}{1}{1}{1}
Partitions with odd multiplicities are counted by
A055922.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],OddQ[Times@@Length/@#]&]],{n,0,8}]
-
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(1/prod(k=1, n, (1 - x^k + O(x*x^n))^u[k])) } \\ Andrew Howroyd, Dec 30 2022
A374688
Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.
Original entry on oeis.org
1, 1, 1, 2, 2, 4, 5, 7, 11, 16, 21, 31, 45, 63, 87, 122, 170, 238, 328, 449, 616, 844, 1151, 1565, 2121, 2861, 3855, 5183, 6953, 9299, 12407, 16513, 21935, 29078, 38468, 50793, 66935, 88037, 115577, 151473, 198175, 258852, 337560, 439507, 571355, 741631
Offset: 0
The a(0) = 1 through a(9) = 16 compositions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9)
(12) (13) (14) (15) (16) (17) (18)
(23) (24) (25) (26) (27)
(122) (123) (34) (35) (36)
(132) (124) (125) (45)
(133) (134) (126)
(142) (143) (135)
(152) (144)
(233) (153)
(1223) (162)
(1232) (234)
(243)
(1224)
(1233)
(1242)
(1323)
Ranked by positions of strictly increasing rows in
A374683 (sums
A374684).
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have
A000041.
- For leaders of anti-runs we have
A374679.
- For leaders of weakly increasing runs we have
A374634.
- For leaders of strictly decreasing runs we have
A374762.
Types of run-leaders (instead of strictly increasing):
- For strictly decreasing leaders we have
A374689.
- For weakly increasing leaders we have
A374690.
- For weakly decreasing leaders we have
A374697.
A374700 counts compositions by sum of leaders of strictly increasing runs.
-
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,Less]&]],{n,0,15}]
Showing 1-10 of 48 results.
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