A100882 -id:A100882 - cp's OEIS frontend

A333190 Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.

1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 26, 29, 39, 49, 50, 68, 80, 92, 109, 129, 142, 181, 201, 227, 262, 317, 343, 404, 456, 516, 589, 677, 742, 870, 949, 1077, 1207, 1385, 1510, 1704, 1895, 2123, 2352, 2649, 2877, 3261, 3571, 3966, 4363, 4873, 5300, 5914, 6466

Offset: 0

Gus Wiseman, May 17 2020

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The non-strict version is A332745.
The generalization to compositions is A333191.
Partitions with distinct run-lengths are A098859.
Partitions with strictly increasing run-lengths are A100471.
Partitions with strictly decreasing run-lengths are A100881.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Partitions with unimodal run-lengths are A332280.
Partitions whose run-lengths are not increasing nor decreasing are A332641.
Compositions whose run-lengths are unimodal or co-unimodal are A332746.
Compositions that are neither increasing nor decreasing are A332834.
Strictly increasing or strictly decreasing compositions are A333147.
Compositions with strictly increasing run-lengths are A333192.
Numbers with strictly increasing prime multiplicities are A334965.
Cf. A032020, A059204, A072706, A332726, A332831, A332833, A332835, A333149.

Table[Length[Select[IntegerPartitions[n],Or[Less@@Length/@Split[#],Greater@@Length/@Split[#]]&]],{n,0,30}]

A229915 Number of espalier polycubes of a given volume in dimension 3.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338

Offset: 0

Matthieu Deneufchâtel, Oct 03 2013

nonn

A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell with coordinates (0,b,c) belongs to the first plateau such that b,c >= 0. If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.

An espalier polycube is a special pyramid such that each plateau contains the cell with coordinates (a,0,0).

Seiichi Manyama, Table of n, a(n) for n = 0..100
C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013.
C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4.

Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.

The generating function for the numbers of espaliers of height h and volumes v_1 , ... v_h is x_1^{n_1} * ... x_h^{n_h} / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).

This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>= ... >= n_h>=1 and then over h.

a(0)=1 prepended by Seiichi Manyama, Aug 20 2020

A316529 Heinz numbers of totally 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, Jul 29 2018

nonn

First differs from A304678 at a(115) = 151, A304678(115) = 150.
The alternating version first differs from this sequence in having 150 and lacking 450.
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.
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.

			Starting with (3,3,2,1), which has Heinz number 150, and repeatedly taking run-lengths gives (3,3,2,1) -> (2,1,1) -> (1,2), so 150 is not in the sequence.
Starting with (3,3,2,2,1), which has Heinz number 450, and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2) -> (1), so 450 is in the sequence.

Cf. A056239, A181819, A182850, A242031, A296150, A305732, A317246.
The enumeration of these partitions by sum is A316496.
The complement is A316597.
The widely normal version is A332291.
The dual version is A335376.
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100883, A133808, A317257, A332275, A332293, A332297.

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

Updated with corrected terminology by Gus Wiseman, Mar 08 2020

A333192 Number of compositions of n with strictly increasing run-lengths.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 16, 24, 31, 37, 51, 67, 76, 103, 129, 158, 199, 242, 293, 370, 450, 538, 652, 799, 953, 1147, 1376, 1635, 1956, 2322, 2757, 3271, 3845, 4539, 5336, 6282, 7366, 8589, 10046, 11735, 13647, 15858, 18442, 21354, 24716, 28630, 32985

Offset: 0

Gus Wiseman, May 17 2020

nonn

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

			The a(1) = 1 through a(8) = 14 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (122)    (33)      (133)      (44)
                    (211)   (311)    (222)     (322)      (233)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (1222)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (11222)
                                               (211111)   (41111)
                                               (1111111)  (122111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, the composition (1,2,2,1,1,1) has run-lengths (1,2,3), so is counted under a(8).

Giovanni Resta, Table of n, a(n) for n = 0..1000

The case of partitions is A100471.
The non-strict version is A332836.
Strictly increasing compositions are A000009.
Unimodal compositions are A001523.
Strict compositions are A032020.
Partitions with strictly increasing run-lengths are A100471.
Partitions with strictly decreasing run-lengths are A100881.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions with strictly increasing or decreasing run-lengths are A333191.
Numbers with strictly increasing prime multiplicities are A334965.
Cf. A072706, A098859, A100882, A100883, A304686, A329744, A329766, A332726, A332833, A332834, A332835, A333147, A333149, A333190.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Length/@Split[#]&]],{n,0,15}]
b[n_, lst_, v_] := b[n, lst, v] = If[n == 0, 1, If[n <= lst, 0, Sum[If[k == v, 0, b[n - k pz, pz, k]], {pz, lst + 1, n}, {k, Floor[n/pz]}]]]; a[n_] := b[n, 0, 0]; a /@ Range[0, 50] (* Giovanni Resta, May 18 2020 *)

Terms a(26) and beyond from Giovanni Resta, May 18 2020

A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 17, 26, 27, 43, 51, 61, 78, 103, 115, 153, 174, 213, 255, 316, 354, 442, 508, 610, 701, 848, 950, 1153, 1303, 1539, 1750, 2075, 2318, 2738, 3081

Offset: 0

Gus Wiseman, Apr 20 2025

			The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (33111)
                          (211111)  (311111)   (51111)
                                    (2111111)  (222111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

For a unique choice we have A000005, ranks A000961.
For at least one choice we have A239455, ranks A351294, conjugate A381432.
For no choices we have A351293, ranks A351295, conjugate A381433.
The complement is A351293 + A000005, ranks too dense.
For equal instead of distinct run-lengths we have A383090, ranks A383089.
These partitions are ranked by A383113 = positions of terms > 1 in A382771.
Cf. A047966, A098859, A130091, A353837, A100471, A100881, A100882, A100883
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A329738 counts compositions with equal run-lengths, ranks A353744.
Cf. A047993, A329739, A381541, A381636, A381717, A382857, A382915, A383013, A383092, A383094, A383097.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], UnsameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]

a(21)-a(38) from Jakub Buczak, May 04 2025

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[#]]]&]

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[#]]]&]

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

A333190 Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.

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A229915 Number of espalier polycubes of a given volume in dimension 3.

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A316529 Heinz numbers of totally strong integer partitions.

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A333192 Number of compositions of n with strictly increasing run-lengths.

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A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

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A335376 Heinz numbers of totally co-strong integer partitions.

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

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