Displaying 1-10 of 11 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 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.
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 totally co-strongly normal integer partitions of n.
+10
10
1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 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
COMMENTS
A sequence of integers is widely totally 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-lengths (co-strong) which are themselves a widely totally co-strongly normal sequence.
Is this sequence bounded?
EXAMPLE
The a(1) = 1 through a(20) = 2 partitions:
1: (1)
2: (11)
3: (21),(111)
4: (211),(1111)
5: (11111)
6: (321),(111111)
7: (1111111)
8: (11111111)
9: (32211),(111111111)
10: (4321),(322111),(1111111111)
11: (11111111111)
12: (111111111111)
13: (1111111111111)
14: (11111111111111)
15: (54321),(111111111111111)
16: (1111111111111111)
17: (11111111111111111)
18: (111111111111111111)
19: (1111111111111111111)
20: (4332221111),(11111111111111111111)
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[IntegerPartitions[n], totnQ]], {n, 0, 30}]
CROSSREFS
Not requiring co-strength gives A332277. The strong version is A332297(n) - 1 for n > 1. The narrow version is a(n) - 1 for n > 1.
The alternating version is A332289. The Heinz numbers of these partitions are A332293. The case of compositions is A332337. Cf. A000009, A100883, A107429, A133808, A181819, A316496, A317245, A317491, A329746, A332279, A332290, A332291, A332292, A332296, A332576.
Number of alternately co-strong reversed integer partitions of n.
+10
9
1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 18, 20, 29, 28, 40, 45, 54, 59, 82, 81, 108, 118, 141, 154, 204, 204, 255, 285, 339, 363, 458, 471, 580, 632, 741, 806, 983, 1015, 1225, 1341, 1562, 1667, 2003, 2107, 2491, 2712, 3101, 3344, 3962, 4182, 4860, 5270, 6022, 6482
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 integer partitions of n.
EXAMPLE
The a(1) = 1 through a(8) = 12 reversed partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (12) (13) (14) (15) (16) (17)
(111) (22) (23) (24) (25) (26)
(1111) (122) (33) (34) (35)
(11111) (123) (124) (44)
(222) (133) (125)
(1122) (1222) (134)
(111111) (1111111) (233)
(1133)
(2222)
(11222)
(11111111)
For example, starting with the composition y = (1,2,3,3,4,4,4) and repeatedly taking run-lengths and reversing gives (1,2,3,3,4,4,4) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2) -> (1). All of these have weakly increasing run-lengths and the last is equal to (1), so y is counted under a(21).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Sort/@IntegerPartitions[n], tniQ]], {n, 0, 30}]
CROSSREFS
The total (instead of alternating) version is A316496. Alternately strong partitions are A317256. The case of ordinary (not reversed) partitions is (also) A317256. The generalization to compositions is A332338.
Number of widely totally normal compositions of n.
+10
8
1, 1, 1, 3, 4, 6, 12, 22, 29, 62, 119, 208, 368, 650, 1197, 2173, 3895, 7022, 12698, 22940, 41564
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.
A composition of n is a finite sequence of positive integers with sum n.
EXAMPLE
The a(1) = 1 through a(7) = 22 compositions:
(1) (11) (12) (112) (122) (123) (1123)
(21) (121) (212) (132) (1132)
(111) (211) (221) (213) (1213)
(1111) (1121) (231) (1231)
(1211) (312) (1312)
(11111) (321) (1321)
(1212) (2113)
(1221) (2122)
(2112) (2131)
(2121) (2212)
(11211) (2311)
(111111) (3112)
(3121)
(3211)
(11221)
(12112)
(12121)
(12211)
(21121)
(111211)
(112111)
(1111111)
For example, starting with y = (3,2,1,1,2,2,2,1,2,1,1,1,1) and repeatedly taking run-lengths gives y -> (1,1,2,3,1,1,4) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1). These are all normal and the last is all 1's, so y is counted under a(20).
MATHEMATICA
recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], recnQ]], {n, 0, 10}]
CROSSREFS
Constantly recursively normal partitions are A332272. The case of reversed partitions is (also) A332277. The co-strong version is (also) A332337. Cf. A001462, A181819, A182850, A317081, A317245, A317491, A329744, A332276, A332289, A332292, A332295, A332297, A332336, A332340.
Heinz numbers of widely alternately co-strongly normal integer partitions.
+10
8
1, 2, 4, 6, 8, 12, 16, 30, 32, 60, 64, 128, 210, 256, 360, 512, 1024, 2048, 2310, 2520, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288
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) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-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 A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.
EXAMPLE
The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
1: ()
2: (1)
4: (1,1)
6: (2,1)
8: (1,1,1)
12: (2,1,1)
16: (1,1,1,1)
30: (3,2,1)
32: (1,1,1,1,1)
60: (3,2,1,1)
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)
360: (3,2,2,1,1,1)
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)
2310: (5,4,3,2,1)
2520: (4,3,2,2,1,1,1)
For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.
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]], LessEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Select[Range[10000], totnQ[Reverse[primeMS[#]]]&]
CROSSREFS
The non-co-strong version is A332276. The enumeration of these partitions by sum is A332289. The total (rather than alternating) version is A332293. Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, 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.
Search completed in 0.009 seconds
|