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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, May 17 2020

Keywords

Examples

			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)

Crossrefs

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.

Programs

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

A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 14, 14, 5, 0, 1, 30, 45, 32, 8, 0, 1, 62, 124, 131, 65, 13, 0, 1, 126, 315, 438, 323, 128, 21, 0, 1, 254, 762, 1305, 1270, 747, 243, 34, 0, 1, 510, 1785, 3612, 4346, 3370, 1629, 452, 55
Offset: 0

Views

Author

Gus Wiseman, Mar 02 2020

Keywords

Comments

In other words, parts of subsequent, non-successive blocks are increasing.

Examples

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    3
    0    1   14   14    5
    0    1   30   45   32    8
    0    1   62  124  131   65   13
    0    1  126  315  438  323  128   21
    0    1  254  762 1305 1270  747  243   34
    ...
Row n = 4 counts the following ordered set partitions:
  {1234}  {1}{234}  {1}{2}{34}  {1}{2}{3}{4}
          {12}{34}  {1}{23}{4}  {1}{2}{4}{3}
          {123}{4}  {12}{3}{4}  {1}{3}{2}{4}
          {124}{3}  {1}{24}{3}  {2}{1}{3}{4}
          {13}{24}  {12}{4}{3}  {2}{1}{4}{3}
          {134}{2}  {1}{3}{24}
          {14}{23}  {13}{2}{4}
          {2}{134}  {1}{34}{2}
          {23}{14}  {1}{4}{23}
          {234}{1}  {2}{1}{34}
          {24}{13}  {2}{13}{4}
          {3}{124}  {2}{14}{3}
          {34}{12}  {23}{1}{4}
          {4}{123}  {3}{12}{4}

Crossrefs

An apparently related triangle is A056242.
Column k = n - 1 is A332724.
Row sums are A332872, which appears to be A007052 shifted right once.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Cf. A072704, A097805, A107429, A227038, A332280, A332283, A332288, A332577.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==k&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,5},{k,0,n}]

A332725 Heinz numbers of integer partitions whose negated first differences are not unimodal.

Original entry on oeis.org

90, 126, 180, 198, 234, 252, 270, 306, 342, 350, 360, 378, 396, 414, 450, 468, 504, 522, 525, 540, 550, 558, 594, 612, 630, 650, 666, 684, 700, 702, 720, 738, 756, 774, 792, 810, 825, 828, 846, 850, 882, 900, 910, 918, 936, 950, 954, 975, 990, 1008, 1026, 1044
Offset: 1

Views

Author

Gus Wiseman, Feb 26 2020

Keywords

Comments

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing 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.

Examples

			The sequence of terms together with their prime indices begins:
    90: {1,2,2,3}
   126: {1,2,2,4}
   180: {1,1,2,2,3}
   198: {1,2,2,5}
   234: {1,2,2,6}
   252: {1,1,2,2,4}
   270: {1,2,2,2,3}
   306: {1,2,2,7}
   342: {1,2,2,8}
   350: {1,3,3,4}
   360: {1,1,1,2,2,3}
   378: {1,2,2,2,4}
   396: {1,1,2,2,5}
   414: {1,2,2,9}
   450: {1,2,2,3,3}
   468: {1,1,2,2,6}
   504: {1,1,1,2,2,4}
   522: {1,2,2,10}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
For example, 350 is the Heinz number of (4,3,3,1), with negated first differences (1,0,2), which is not unimodal, so 350 is in the sequence.

Links

Crossrefs

The complement is too full.
The enumeration of these partitions by sum is A332284.
The version where the last part is taken to be 0 is A332832.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Heinz numbers of partitions with weakly increasing differences are A325360.
Cf. A001523, A007052, A240026, A332280, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Select[Range[1000],!unimodQ[Differences[primeMS[#]]]&]
Previous Showing 31-33 of 33 results.

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