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Number of totally strong integer partitions of n.
+10
20
1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668
COMMENTS
An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.
EXAMPLE
The a(1) = 1 through a(8) = 12 totally strong partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(321) (421) (332)
(2211) (2221) (431)
(111111) (1111111) (521)
(2222)
(3311)
(22211)
(11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.
MATHEMATICA
totincQ[q_]:=Or[q=={}, q=={1}, And[GreaterEqual@@Length/@Split[q], totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n], totincQ]], {n, 0, 30}]
CROSSREFS
The Heinz numbers of these partitions are A316529. The version for compositions is A332274. The version for reversed partitions is (also) A332275. The narrowly normal version is A332297. Partitions with weakly decreasing run-lengths are A100882.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 07 2020
Number of widely alternately strongly normal integer partitions of n.
+10
15
1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1
COMMENTS
An integer partition is widely alternately strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which, if reversed, are themselves a widely alternately strongly normal partition.
Also the number of widely alternately co-strongly normal reversed integer partitions of n.
EXAMPLE
The a(1) = 1, a(3) = 2, and a(21) = 3 partitions:
(1) (21) (654321)
(111) (4443321)
(111111111111111111111)
For example, starting with the partition y = (4,4,4,3,3,2,1) and repeatedly taking run-lengths and reversing gives (4,4,4,3,3,2,1) -> (1,1,2,3) -> (1,1,2) -> (1,2) -> (1,1). All of these are normal with weakly decreasing run-lengths, and the last is all 1's, so y is counted under a(21).
MATHEMATICA
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], GreaterEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[IntegerPartitions[n], totnQ]], {n, 0, 30}]
CROSSREFS
The case of reversed partitions is (also) A332289. The case of compositions is A332340. Cf. A100883, A181819, A317081, A317245, A317256, A317491, A329746, A329747, A332278, A332290, A332291, A332297, A332337.
Number of widely alternately co-strongly normal compositions of n.
+10
15
1, 1, 1, 3, 3, 4, 9, 11, 13, 23, 53, 78, 120, 207, 357, 707, 1183, 2030, 3558, 6229, 10868
COMMENTS
An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
EXAMPLE
The a(1) = 1 through a(8) = 13 compositions:
(1) (11) (12) (121) (122) (123) (1213) (1232)
(21) (211) (212) (132) (1231) (1322)
(111) (1111) (1211) (213) (1312) (2123)
(11111) (231) (1321) (2132)
(312) (2122) (2312)
(321) (2131) (2321)
(1212) (2311) (3122)
(2121) (3121) (3212)
(111111) (3211) (12131)
(12121) (13121)
(1111111) (21212)
(122111)
(11111111)
For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).
MATHEMATICA
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], totnQ]], {n, 0, 10}]
CROSSREFS
Compositions with normal run-lengths are A329766. The Heinz numbers of the case of partitions are A332290. The total (instead of alternating) version is A332337. Not requiring normality gives A332338. The strong version is this same sequence.
The narrow version is a(n) + 1 for n > 1.
Number of widely totally normal integer partitions of n.
+10
14
1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 2, 4, 4, 6, 3, 5, 7, 6, 8, 12, 9, 12, 13, 11, 12, 18, 17, 12, 32, 19, 25, 33, 30, 28, 44, 33, 43, 57, 51, 60, 83, 70, 83, 103, 96, 97, 125, 117, 134, 157, 157, 171, 226, 215, 238, 278, 302, 312, 359, 357, 396, 450, 444, 477, 580
COMMENTS
A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
Also the number of widely totally normal reversed integer partitions of n.
EXAMPLE
The a(n) partitions for n = 1, 4, 10, 11, 16, 18:
1 211 4321 33221 443221 543321
1111 33211 322211 4432111 4333221
322111 332111 1111111111111111 4432221
1111111111 11111111111 4433211
43322211
44322111
111111111111111111
MATHEMATICA
recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n], recnQ]], {n, 0, 30}]
CROSSREFS
Taking multiplicities instead of run-lengths gives A317245. Constantly recursively normal partitions are A332272. The Heinz numbers of these partitions are A332276. The case of all compositions (not just partitions) is A332279. The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.
Heinz numbers of widely totally strongly normal integer partitions.
+10
13
1, 2, 4, 6, 8, 16, 18, 30, 32, 64, 128, 210, 256, 450, 512, 1024, 2048, 2250, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 131072, 262144, 510510, 524288
COMMENTS
An integer partition is widely totally strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which are themselves a widely totally strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A304660, so there are infinitely many terms that are not powers of 2 or primorial numbers.
EXAMPLE
The sequence of all widely totally strongly normal integer partitions together with their Heinz numbers begins:
1: ()
2: (1)
4: (1,1)
6: (2,1)
8: (1,1,1)
16: (1,1,1,1)
18: (2,2,1)
30: (3,2,1)
32: (1,1,1,1,1)
64: (1,1,1,1,1,1)
128: (1,1,1,1,1,1,1)
210: (4,3,2,1)
256: (1,1,1,1,1,1,1,1)
450: (3,3,2,2,1)
512: (1,1,1,1,1,1,1,1,1)
1024: (1,1,1,1,1,1,1,1,1,1)
2048: (1,1,1,1,1,1,1,1,1,1,1)
2250: (3,3,3,2,2,1)
2310: (5,4,3,2,1)
4096: (1,1,1,1,1,1,1,1,1,1,1,1)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], GreaterEqual@@Length/@Split[ptn], totnQ[Length/@Split[ptn]]]];
Select[Range[10000], totnQ[Reverse[primeMS[#]]]&]
CROSSREFS
The case of reversed partitions is (also) A332293. Heinz numbers of normal partitions with decreasing run-lengths are A025487. Cf. A055932, A056239, A181819, A242031, A317089, A317246, A317257, A317492, A329747, A332277, A332278, A332290, A332292, A332297, A332337.
Number of narrowly totally strongly normal integer partitions of n.
+10
12
1, 1, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2
COMMENTS
A partition is narrowly totally strongly normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a narrowly totally strongly normal partition.
EXAMPLE
The a(1) = 1, a(2) = 2, a(3) = 3, and a(55) = 4 partitions:
(1) (2) (3) (55)
(1,1) (2,1) (10,9,8,7,6,5,4,3,2,1)
(1,1,1) (5,5,5,5,5,4,4,4,4,3,3,3,2,2,1)
(1)^55
For example, starting with the partition (3,3,2,2,1) and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2). The first four are normal and have weakly decreasing run-lengths, and the last is a singleton, so (3,3,2,2,1) is counted under a(11).
MATHEMATICA
tinQ[q_]:=Or[q=={}, Length[q]==1, And[Union[q]==Range[Max[q]], GreaterEqual@@Length/@Split[q], tinQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n], tinQ]], {n, 0, 30}]
CROSSREFS
The non-totally normal version is A316496. The widely alternating version is A332292. The non-strong case of compositions is A332296. The case of compositions is A332336. The wide version is a(n) - 1 for n > 1.
Cf. A001462, A025487, A100883, A181819, A182850, A317081, A317245, A317256, A317491, A332275, A332277, A332278, A332291, A332337.
Number of widely totally strongly normal compositions of n.
+10
12
1, 1, 1, 3, 3, 3, 9, 9, 12, 23, 54, 77, 116, 205, 352, 697, 1174, 2013, 3538, 6209, 10830
COMMENTS
A sequence is widely totally strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a widely totally strongly normal sequence.
EXAMPLE
The a(1) = 1 through a(8) = 12 compositions:
(1) (11) (12) (112) (212) (123) (1213) (1232)
(21) (121) (221) (132) (1231) (2123)
(111) (1111) (11111) (213) (1312) (2132)
(231) (1321) (2312)
(312) (2131) (2321)
(321) (3121) (3212)
(1212) (11221) (12131)
(2121) (12121) (13121)
(111111) (1111111) (21212)
(22112)
(111221)
(11111111)
For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11). These are all normal with weakly decreasing run-lengths, and the last is all 1's, so (22112) is counted under a(8).
MATHEMATICA
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], totnQ]], {n, 0, 10}]
CROSSREFS
Heinz numbers in the case of partitions are A332291. The alternating version is A332340. The co-strong version is this same sequence.
Cf. A025487, A100883, A181819, A317245, A317491, A329744, A332274, A332276, A332277, A332292, A332293, A332296.
Number of widely alternately co-strongly normal integer partitions of n.
+10
11
1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1
COMMENTS
An integer partition is widely alternately co-strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
EXAMPLE
The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:
(1) (21) (4321)
(111) (322111)
(1111111111)
For example, starting with y = (4,3,2,2,1,1,1) and repeatedly taking run-lengths and reversing gives y -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal, have weakly increasing run-lengths, and the last is all 1's, so y is counted a(14).
MATHEMATICA
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[IntegerPartitions[n], totnQ]], {n, 0, 30}]
CROSSREFS
The non-co-strong version is A332277. The total (instead of alternate) version is A332278. The Heinz numbers of these partitions are A332290. The case of reversed partitions is (also) A332292. The generalization to compositions is A332340. Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339.
Number of narrowly totally normal compositions of n.
+10
9
1, 1, 2, 4, 5, 7, 13, 23, 30, 63, 120, 209, 369, 651, 1198, 2174, 3896, 7023, 12699, 22941, 41565
COMMENTS
A sequence is narrowly totally normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) with narrowly totally normal run-lengths.
A composition of n is a finite sequence of positive integers summing to n.
EXAMPLE
The a(0) = 1 through a(6) = 13 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (112) (122) (123)
(21) (121) (212) (132)
(111) (211) (221) (213)
(1111) (1121) (231)
(1211) (312)
(11111) (321)
(1212)
(1221)
(2112)
(2121)
(11211)
(111111)
For example, starting with the composition (1,1,2,3,1,1) and repeatedly taking run-lengths gives (1,1,2,3,1,1) -> (2,1,1,2) -> (1,2,1) -> (1,1,1) -> (3). The first four are normal and the last is a singleton, so (1,1,2,3,1,1) is counted under a(9).
MATHEMATICA
tinQ[q_]:=Or[Length[q]<=1, And[Union[q]==Range[Max[q]], tinQ[Length/@Split[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], tinQ]], {n, 0, 10}]
CROSSREFS
The wide recursive version (for partitions) is A332295. The alternating version is A332296 (this sequence). The co-strong version is (also) A332336. Cf. A001462, A316496, A317081, A317245, A317491, A329744, A332276, A332277, A332278, A332297, A332337, A332340.
Heinz numbers of widely totally co-strongly normal integer partitions.
+10
7
1, 2, 4, 6, 8, 12, 16, 30, 32, 64, 128, 180, 210, 256, 360, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288
COMMENTS
An integer partition is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
30: {1,2,3}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
128: {1,1,1,1,1,1,1}
180: {1,1,2,2,3}
210: {1,2,3,4}
256: {1,1,1,1,1,1,1,1}
360: {1,1,1,2,2,3}
512: {1,1,1,1,1,1,1,1,1}
1024: {1,1,1,1,1,1,1,1,1,1}
2048: {1,1,1,1,1,1,1,1,1,1,1}
2310: {1,2,3,4,5}
4096: {1,1,1,1,1,1,1,1,1,1,1,1}
8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
gnaQ[y_]:=Or[y=={}, Union[y]=={1}, And[normQ[y], LessEqual@@Length/@Split[y], gnaQ[Length/@Split[y]]]];
Select[Range[1000], gnaQ[Reverse[primeMS[#]]]&]
CROSSREFS
The non-co-strong version is A332276. The enumeration of these partitions by sum is A332278. The alternating version is A332290. The case of reversed partitions is (also) A332291. Cf. A000009, A056239, A133808, A182850, A304660, A317089, A317246, A317257, A317492, A329747, A332277, A332289.
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