Displaying 1-10 of 15 results found.
Number of alternately co-strong integer partitions of n.
+10
16
1, 1, 2, 3, 5, 6, 11, 13, 19, 25, 35, 42, 61, 74, 98, 122, 161, 194, 254, 304, 388, 472, 589, 700, 878, 1044, 1278, 1525, 1851, 2182, 2651, 3113, 3735, 4389, 5231, 6106, 7278, 8464, 9995, 11631, 13680, 15831, 18602, 21463, 25068, 28927, 33654, 38671, 44942, 51514
COMMENTS
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
Also the number of alternately strong reversed integer partitions of n.
EXAMPLE
The a(1) = 1 through a(7) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(211) (311) (51) (61)
(1111) (2111) (222) (322)
(11111) (321) (421)
(411) (511)
(2211) (3211)
(3111) (4111)
(21111) (22111)
(111111) (31111)
(211111)
(1111111)
For example, starting with the partition y = (3,2,2,1,1) and repeatedly taking run-lengths and reversing gives (3,2,2,1,1) -> (2,2,1) -> (1,2), which is not weakly decreasing, so y is not alternately co-strong. On the other hand, we have (3,3,2,2,1,1,1) -> (3,2,2) -> (2,1) -> (1,1) -> (2) -> (1), so (3,3,2,2,1,1,1) is counted under a(13).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[IntegerPartitions[n], tniQ]], {n, 0, 30}]
CROSSREFS
Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A317081, A317086, A317245, A317258. The Heinz numbers of these partitions are given by A317257. The total (instead of alternating) version is A332275. Dominates A332289 (the normal version). The generalization to compositions is A332338. The case of reversed partitions is (also) A332339.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 08 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 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 alternately co-strong integer partitions.
+10
13
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70
COMMENTS
The first term absent from this sequence but present in A242031 is 180. A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
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: {} 16: {1,1,1,1} 32: {1,1,1,1,1}
2: {1} 17: {7} 33: {2,5}
3: {2} 19: {8} 34: {1,7}
4: {1,1} 20: {1,1,3} 35: {3,4}
5: {3} 21: {2,4} 36: {1,1,2,2}
6: {1,2} 22: {1,5} 37: {12}
7: {4} 23: {9} 38: {1,8}
8: {1,1,1} 24: {1,1,1,2} 39: {2,6}
9: {2,2} 25: {3,3} 40: {1,1,1,3}
10: {1,3} 26: {1,6} 41: {13}
11: {5} 27: {2,2,2} 42: {1,2,4}
12: {1,1,2} 28: {1,1,4} 43: {14}
13: {6} 29: {10} 44: {1,1,5}
14: {1,4} 30: {1,2,3} 45: {2,2,3}
15: {2,3} 31: {11} 46: {1,9}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totincQ[q_]:=Or[Length[q]<=1, And[OrderedQ[Length/@Split[q]], totincQ[Reverse[Length/@Split[q]]]]];
Select[Range[100], totincQ[Reverse[primeMS[#]]]&]
CROSSREFS
These partitions are counted by A317256. Totally co-strong partitions are counted by A332275. Alternately co-strong compositions are counted by A332338. Alternately co-strong reversed partitions are counted by A332339.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Jun 04 2020
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 totally co-strong integer partitions of n.
+10
11
1, 1, 2, 3, 5, 6, 11, 12, 17, 22, 30, 32, 49, 53, 70, 82, 108, 119, 156, 171, 219, 250, 305, 336, 424, 468, 562, 637, 754, 835, 1011, 1108, 1304, 1461, 1692, 1873, 2212, 2417, 2787, 3109, 3562, 3911, 4536, 4947, 5653, 6265, 7076, 7758, 8883, 9669, 10945, 12040
COMMENTS
A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.
Also the number of totally strong reversed integer partitions of n.
EXAMPLE
The a(1) = 1 through a(7) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(211) (311) (51) (61)
(1111) (2111) (222) (322)
(11111) (321) (421)
(411) (511)
(2211) (4111)
(3111) (22111)
(21111) (31111)
(111111) (211111)
(1111111)
For example, the partition y = (5,4,4,4,3,3,3,2,2,2,2,2,2,1,1,1,1,1,1) has run-lengths (1,3,3,6,6), with run-lengths (1,2,2), with run-lengths (1,2), with run-lengths (1,1), with run-lengths (2), with run-lengths (1). All of these having weakly increasing run-lengths, and the last is (1), so y is counted under a(44).
MATHEMATICA
totincQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n], totincQ]], {n, 0, 30}]
CROSSREFS
The version for reversed partitions is (also) A316496. The alternating version is A317256. The generalization to compositions is A332274. Cf. A001462, A100883, A181819, A182850, A317491, A329746, A332289, A332297, A332336, A332337, A332338, A332339, A332340.
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.
Number of narrowly totally strongly normal compositions of n.
+10
9
1, 1, 2, 4, 4, 4, 10, 10, 13, 24, 55, 78, 117, 206, 353, 698, 1175, 2014, 3539, 6210, 10831
COMMENTS
A sequence 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 sequence.
A composition of n is a finite sequence of positive integers summing to n.
EXAMPLE
The a(1) = 1 through a(8) = 13 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(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) -> (2). The first four are normal with weakly decreasing run-lengths, and the last is a singleton, so (22112) is counted under a(8).
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[Join@@Permutations/@IntegerPartitions[n], tinQ]], {n, 0, 10}]
CROSSREFS
The co-strong version is A332336 (this sequence). Cf. A025487, A316496, A317081, A317245, A317256, A317491, A329744, A332279, A332291, A332292, A332338, A332340.
Number of alternately co-strong compositions of n.
+10
9
1, 1, 2, 4, 7, 12, 24, 39, 72, 125, 224, 387, 697, 1205, 2141, 3736, 6598, 11516, 20331, 35526, 62507, 109436, 192200, 336533, 590582, 1034187
COMMENTS
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
EXAMPLE
The a(1) = 1 through a(5) = 12 compositions:
(1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(121) (113)
(1111) (131)
(212)
(221)
(1112)
(1121)
(11111)
For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], tniQ]], {n, 0, 10}]
CROSSREFS
The recursive (rather than alternating) version is A332274. The total (rather than alternating) version is (also) A332274. The strong version is this same sequence.
The case of reversed partitions is A332339. The normal version is A332340(n) + 1 for n > 1. Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.
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