A317258 -id:A317258 - cp's OEIS frontend

A317256 Number of alternately co-strong integer partitions of n.

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

Offset: 0

Gus Wiseman, Jul 25 2018

nonn

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.

			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).

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 dual version is A332339.
The case of reversed partitions is (also) A332339.
Cf. A316496  A317491, A329746, A332292, A332297, A332340.

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}]

Updated with corrected terminology by Gus Wiseman, Mar 08 2020

A317257 Heinz numbers of alternately co-strong integer partitions.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 25 2018

nonn

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).

			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}

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A056239, A100883, A181819, A182850, A242031, A296150, A305732, A317246.
These partitions are counted by A317256.
The complement is A317258.
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.
The total version is A335376.
Cf. A182857, A304660, A305563, A316496, A332292, A332340.

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[#]]]&]

Updated with corrected terminology by Gus Wiseman, Jun 04 2020

A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

The first term of this sequence that is absent from A112769 is 150.

An integer partition is totally nondecreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly increasing and, taken in reverse order, are themselves a totally nondecreasing integer partition.

			150 is the Heinz number of (3,3,2,1), with multiplicities (1,1,2), which has multiplicities (2,1), which are decreasing, so 150 does not belong to the sequence.

Complement of A316529.

Cf. A056239, A071365, A112769, A181819, A182850, A242031, A296150, A305733, A316496, A317258.

A335376 Heinz numbers of totally co-strong integer partitions.

Original entry on oeis.org

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, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jun 04 2020

nonn

First differs from A242031 and A317257 in lacking 60.
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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			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}
For example, 180 is the Heinz number of (3,2,2,1,1) which has run-lengths: (1,2,2) -> (1,2) -> (1,1) -> (2) -> (1). All of these are weakly increasing, so 180 is in the sequence.

Partitions with weakly increasing run-lengths are A100883.
Totally strong partitions are counted by A316496.
The strong version is A316529.
The version for reversed partitions is (also) A316529.
These partitions are counted by A332275.
The widely normal version is A332293.
The complement is A335377.
Cf. A100882, A133808, A181819, A182850, A242031, A305732, A317256, A317258, A329747, A332291.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];
Select[Range[100],totcostrQ[Reverse[primeMS[#]]]&]

A335377 Heinz numbers of non-totally co-strong integer partitions.

Original entry on oeis.org

18, 50, 54, 60, 75, 84, 90, 98, 108, 120, 126, 132, 140, 147, 150, 156, 162, 168, 198, 204, 220, 228, 234, 240, 242, 245, 250, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 324, 336, 338, 340, 342, 348, 350, 363, 364, 372, 375, 378, 380, 408, 414, 420

Offset: 1

Gus Wiseman, Jun 05 2020

nonn

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   18: {1,2,2}        156: {1,1,2,6}        276: {1,1,2,9}
   50: {1,3,3}        162: {1,2,2,2,2}      280: {1,1,1,3,4}
   54: {1,2,2,2}      168: {1,1,1,2,4}      294: {1,2,4,4}
   60: {1,1,2,3}      198: {1,2,2,5}        300: {1,1,2,3,3}
   75: {2,3,3}        204: {1,1,2,7}        306: {1,2,2,7}
   84: {1,1,2,4}      220: {1,1,3,5}        308: {1,1,4,5}
   90: {1,2,2,3}      228: {1,1,2,8}        312: {1,1,1,2,6}
   98: {1,4,4}        234: {1,2,2,6}        315: {2,2,3,4}
  108: {1,1,2,2,2}    240: {1,1,1,1,2,3}    324: {1,1,2,2,2,2}
  120: {1,1,1,2,3}    242: {1,5,5}          336: {1,1,1,1,2,4}
  126: {1,2,2,4}      245: {3,4,4}          338: {1,6,6}
  132: {1,1,2,5}      250: {1,3,3,3}        340: {1,1,3,7}
  140: {1,1,3,4}      260: {1,1,3,6}        342: {1,2,2,8}
  147: {2,4,4}        264: {1,1,1,2,5}      348: {1,1,2,10}
  150: {1,2,3,3}      270: {1,2,2,2,3}      350: {1,3,3,4}
For example, 60 is the Heinz number of (3,2,1,1), which has run-lengths: (1,1,2) -> (2,1) -> (1,1) -> (2) -> (1). Since (2,1) is not weakly increasing, 60 is in the sequence.

Partitions with weakly increasing run-lengths are counted by A100883.
Totally strong partitions are counted by A316496.
Heinz numbers of totally strong partitions are A316529.
The version for reversed partitions is A316597.
The strong version is (also) A316597.
The alternating version is A317258.
Totally co-strong partitions are counted by A332275.
The complement is A335376.
Cf. A100882, A181819, A182850, A242031, A305732, A317256, A329747, A332291, A332293, A332339.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];
Select[Range[100],!totcostrQ[Reverse[primeMS[#]]]&]

cp's OEIS Frontend

A317256 Number of alternately co-strong integer partitions of n.

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A317257 Heinz numbers of alternately co-strong integer partitions.

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A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

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A335376 Heinz numbers of totally co-strong integer partitions.

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A335377 Heinz numbers of non-totally co-strong integer partitions.

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