A317084
Number of integer partitions of n whose multiplicities are weakly increasing and span an initial interval of positive integers.
Original entry on oeis.org
1, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 17, 23, 27, 34, 38, 50, 54, 70, 79, 97, 107, 135, 148, 180, 205, 243, 270, 328, 360, 429, 480, 561, 625, 738, 810, 949, 1057, 1219, 1349, 1571, 1723, 1986, 2206, 2515, 2776, 3188, 3496, 3983, 4408, 4980, 5485, 6228, 6826
Offset: 0
The a(7) = 6 integer partitions are (7), (61), (52), (43), (421), (331).
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normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[Length/@Split[#]],OrderedQ[Reverse[Length/@Split[#]]]]&]],{n,60}]
A335941
Number of partitions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.
Original entry on oeis.org
1, 1, 2, 1, 1, 4, 2, 5, 5, 9, 8, 15, 11, 14, 22, 28, 30, 36, 37, 53, 60, 80, 83, 104, 114, 148, 157, 201, 218, 283, 284, 362, 400, 455, 518, 624, 697, 807, 907, 1036, 1181, 1368, 1531, 1727, 1990, 2197, 2563, 2849, 3182, 3568, 4095, 4548, 5143, 5720, 6420
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 2: 11, 2.
a(3) = 1: 21.
a(4) = 1: 211.
a(5) = 4: 2111, 221, 311, 32.
a(6) = 2: 2211, 321.
a(7) = 5: 22111, 2221, 3211, 322, 331.
a(8) = 5: 22211, 32111, 3221, 3311, 332.
a(9) = 9: 222111, 321111, 32211, 3222, 33111, 3321, 42111, 4311, 432.
a(10) = 8: 2221111, 322111, 32221, 331111, 33211, 4222, 4321, 433.
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b:= proc(n,i,s) option remember;
`if`(n=0, `if`(s={$0..max(s)}, 1, 0), `if`(i<1, 0, add(
b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}), j=0..n/i)))
end:
a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}):
seq(a(n), n=0..55);
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b[n_, i_, s_] := b[n, i, s] =
If[n == 0, If[s == Range[0, Max[s]], 1, 0], If[i < 1, 0, Sum[
b[n-i*j, i-1, Union@Flatten@{s, j, If[j == 0, 0, i]}], {j, 0, n/i}]]];
a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)
A330937
Number of strictly recursively normal integer partitions of n.
Original entry on oeis.org
1, 2, 3, 5, 7, 10, 15, 20, 27, 35, 49, 58, 81, 100, 126, 160, 206, 246, 316, 374, 462, 564, 696, 813, 1006, 1195, 1441, 1701, 2058, 2394, 2896, 3367, 4007, 4670, 5542, 6368, 7540, 8702, 10199, 11734, 13760, 15734, 18384, 21008, 24441, 27893, 32380, 36841
Offset: 0
The a(1) = 1 through a(9) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(211) (41) (51) (52) (62) (63)
(221) (321) (61) (71) (72)
(311) (411) (322) (332) (81)
(331) (422) (432)
(421) (431) (441)
(511) (521) (522)
(3211) (611) (531)
(3221) (621)
(4211) (711)
(3321)
(4221)
(4311)
(5211)
(32211)
The narrow instead of strict version is
A332272.
A wide instead of strict version is
A332295(n) - 1 for n > 1.
Cf.
A107429,
A181819,
A316496,
A317081,
A317245,
A317491,
A329744,
A329746,
A329766,
A332277,
A332576.
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normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[ptn=={},UnsameQ@@qtn,And[normQ[qtn],recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]
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