A100883 -id:A100883 - cp's OEIS frontend

A244395 Number of partitions of n in which the largest summand has frequency 1, the next largest summand has frequency at most 2, the third largest has frequency at most 3, etc.

1, 1, 1, 2, 3, 4, 5, 8, 11, 15, 20, 26, 34, 46, 59, 78, 101, 129, 163, 209, 261, 329, 412, 517, 641, 798, 986, 1216, 1493, 1829, 2229, 2721, 3303, 4000, 4841, 5841, 7034, 8458, 10144, 12137, 14512, 17306, 20596, 24483, 29045, 34391, 40680, 48032, 56627, 66666

Offset: 0

David S. Newman, Jul 03 2014

nonn

			For n=6 the partitions counted are: 6, 51, 42, 411, 321.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A100471, A100883, A244393, A295261.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, t) +add(b(n-i*j, i-1, t+1), j=1..min(t, n/i))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

nend = 20;
For[n = 1, n <= nend, n++,
count[n] = 0;
Ip = IntegerPartitions[n];
For[i = 1, i <= Length[Ip], i++,
m = Max[Ip[[i]]];
condition = True;
Tip = Tally[Ip[[i]]];
For[j = 1, j <= Length[Tip], j++,
condition = condition && (Tip[[j]][[2]] <= j)];
If[condition, count[n]++ (* ; Print[Ip[[i]]] *)]];
]
Table[count[i], {i, 1, nend}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0,
    b[n, i-1, t] + Sum[b[n-i*j, i-1, t+1], {j, 1, Min[t, n/i]}]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 03 2014

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[#]]]&]

A332871 Number of compositions of n whose run-lengths are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 8, 24, 55, 128, 282, 625, 1336, 2855, 6000, 12551, 26022, 53744, 110361, 225914, 460756, 937413, 1902370, 3853445, 7791647, 15732468, 31725191, 63907437, 128613224, 258626480, 519700800, 1043690354, 2094882574, 4202903667, 8428794336, 16897836060

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

Also compositions whose run-lengths are not weakly decreasing.

			The a(4) = 1 through a(6) = 8 compositions:
  (112)  (113)   (114)
         (221)   (1113)
         (1112)  (1131)
         (1121)  (1221)
                 (2112)
                 (11112)
                 (11121)
                 (11211)
For example, the composition (2,1,1,2) has run-lengths (1,2,1), which are not weakly increasing, so (2,1,1,2) is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for the compositions themselves (not run-lengths) is A056823.
The version for unsorted prime signature is A112769, with dual A071365.
The case without weakly decreasing run-lengths either is A332833.
The complement is counted by A332836.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are not unimodal are A332727.
Cf. A001523, A072704, A100883, A181819, A329744, A329766, A332641, A332669, A332726, A332745, A332746, A332834, A332835.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!LessEqual@@Length/@Split[#]&]],{n,0,10}]

a(n) = 2^(n - 1) - A332836(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A296116 Number of partitions in which each summand, s, may be used with frequency f if f divides s.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 6, 9, 12, 14, 18, 23, 29, 35, 43, 56, 68, 82, 100, 122, 147, 174, 209, 252, 302, 356, 421, 500, 589, 690, 808, 952, 1110, 1292, 1505, 1756, 2034, 2348, 2715, 3139, 3620, 4156, 4778, 5492, 6296, 7195, 8220, 9398, 10714, 12194, 13872, 15784

Offset: 0

David S. Newman, Dec 04 2017

nonn

			For n=3, the partitions counted are 3 and 2+1.
For n=4: 4, 3+1, 2+2.
For n=5: 5, 4+1, 3+2, 2+2+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A100471, A100881, A100882, A100883.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or n<0, 0,
      b(n, i-1)+add(b(n-i*j, i-1), j=numtheory[divisors](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 05 2017

iend = 30;
s = Series[Product[1 + Sum[x^(Divisors[n][[i]] n), {i, 1, Length[Divisors[n]]}], {n, 1, iend}], {x, 0, iend}]; Print[s];
CoefficientList[s, x]

G.f.: Product_{n >= 1} (1 + Sum_{d divides n} x^(d*n)).

More terms from Alois P. Heinz, Dec 05 2017

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[#]]]&]

A334969 Heinz numbers of alternately strong integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Jun 09 2020

nonn

First differs from A304678 in lacking 450.
First differs from A316529 (the totally strong version) in having 150.
A sequence is alternately strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and, when reversed, are themselves an alternately 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 does not contain 450, the Heinz number of (3,3,2,2,1), because, while the multiplicities are weakly decreasing, their reverse (1,2,2) does not have weakly decreasing multiplicities.

The co-strong version is A317257.
The case of reversed partitions is (also) A317257.
The total version is A316529.
These partitions are counted by A332339.
Totally co-strong partitions are counted by A332275.
Alternately co-strong compositions are counted by A332338.
Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.

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

A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Gus Wiseman, Apr 17 2025

nonn

First differs from A381871 in having 36.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 36 are {1,1,2,2}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}

For run-lengths instead of sums we have A059404, distinct A130092.
The complement is A353833, counted by A304442.
For distinct instead of equal run-sums we have A353839.
Partitions of this type are counted by A382076.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with a common run-sum, ranks A353848.
A353862 gives the greatest run-sum of prime indices, least A353931.
A382877 counts permutations of prime indices with equal run-sums, zeros A383100.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.

Select[Range[100], !SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

cp's OEIS Frontend

A244395 Number of partitions of n in which the largest summand has frequency 1, the next largest summand has frequency at most 2, the third largest has frequency at most 3, etc.

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A335376 Heinz numbers of totally co-strong integer partitions.

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A332871 Number of compositions of n whose run-lengths are not weakly increasing.

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A296116 Number of partitions in which each summand, s, may be used with frequency f if f divides s.

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

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A334969 Heinz numbers of alternately strong integer partitions.

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A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

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