A358905
Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.
Original entry on oeis.org
1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111
Offset: 0
The a(0) = 1 through a(4) = 13 sequences:
() ((1)) ((2)) ((3)) ((4))
((11)) ((21)) ((22))
((1)(1)) ((111)) ((31))
((1)(2)) ((211))
((2)(1)) ((1111))
((1)(1)(1)) ((1)(3))
((2)(2))
((3)(1))
((11)(11))
((1)(1)(2))
((1)(2)(1))
((2)(1)(1))
((1)(1)(1)(1))
The case of set partitions is
A038041.
The version for weakly decreasing lengths is
A141199, strictly
A358836.
For equal sums instead of lengths we have
A279787.
The case of plane partitions is
A323429.
The case of constant sums also is
A358833.
A055887 counts sequences of partitions with total sum n.
-
ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]
-
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(1 + sum(k=1, n, 1/(1 - polcoef(g, k, y)) - 1))} \\ Andrew Howroyd, Dec 31 2022
A358833
Number of rectangular twice-partitions of n of type (P,R,P).
Original entry on oeis.org
1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150
Offset: 0
The a(1) = 1 through a(5) = 8 twice-partitions:
(1) (2) (3) (4) (5)
(11) (21) (22) (32)
(1)(1) (111) (31) (41)
(1)(1)(1) (211) (221)
(1111) (311)
(2)(2) (2111)
(11)(11) (11111)
(1)(1)(1)(1) (1)(1)(1)(1)(1)
This is the rectangular case of
A279787.
This is the case of
A306319 with constant sums.
For distinct instead of constant lengths and sums we have
A358832.
The version for multiset partitions of integer partitions is
A358835.
-
twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{n,0,10}]
-
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, polcoef(p, j, y)^d))))} \\ Andrew Howroyd, Dec 31 2022
A374704
Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.
Original entry on oeis.org
1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075
Offset: 0
The a(0) = 1 through a(4) = 15 ways:
() ((1)) ((2)) ((3)) ((4))
((1,1)) ((1,2)) ((1,3))
((1),(1)) ((1,1,1)) ((2,2))
((1),(1,1)) ((1,1,2))
((1,1),(1)) ((2),(2))
((1),(1),(1)) ((1,1,1,1))
((1),(1,2))
((1,2),(1))
((1),(1,1,1))
((1,1),(1,1))
((1,1,1),(1))
((1),(1),(1,1))
((1),(1,1),(1))
((1,1),(1),(1))
((1),(1),(1),(1))
A variation for weakly increasing lengths is
A141199.
For identical sums instead of minima we have
A279787.
For maxima instead of minima, or for unreversed partitions, we have
A358905.
A055887 counts sequences of partitions with total sum n.
Cf.
A000041,
A063834,
A106356,
A189076,
A238343,
A304969,
A305551,
A319066,
A323429,
A333213,
A358833,
A358835.
-
Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]
-
seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024
Showing 1-3 of 3 results.
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