cp's OEIS Frontend

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.

Previous Showing 11-17 of 17 results.

A332746 Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 134, 174, 227, 291, 373, 473, 598, 748, 936, 1163, 1437, 1771, 2170, 2651, 3226, 3916, 4727, 5702, 6846, 8205, 9793, 11681, 13866, 16462, 19452, 22976, 27041, 31820, 37276, 43693, 51023, 59559, 69309, 80664
Offset: 0

Views

Author

Gus Wiseman, Feb 27 2020

Keywords

Comments

First differs from A000041 at a(14) = 134, A000041(14) = 135.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

			The only partition not counted under a(14) = 134 is (4,3,3,2,1,1), whose run-lengths (1,2,1,2) are neither unimodal nor is their negation.

Links

Crossrefs

Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The complement is counted by A332640.
The Heinz numbers of partitions not in this class are A332643.
Unimodal compositions are A001523.
Partitions with unimodal run-lengths are A332280.
Compositions whose negation is unimodal are A332578.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths are neither weakly increasing nor weakly decreasing: A332641.
Run-lengths and negated run-lengths are both unimodal: A332745.
Cf. A007052, A025065, A100883, A115981, A181819, A332283, A332577, A332578, A332642, A332669, A332726, A332831.

Programs

  • Mathematica
    unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],unimodQ[Length/@Split[#]]||unimodQ[-Length/@Split[#]]&]],{n,0,30}]

A332728 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 13, 14, 17, 22, 24, 28, 34, 37, 43, 53, 56, 64, 76, 83, 93, 111, 117, 131, 153, 163, 182, 210, 225, 250, 284, 304, 332, 377, 401, 441, 497, 529, 576, 647, 687, 745, 830, 883, 955, 1062, 1127, 1216, 1339, 1422, 1532, 1684, 1779, 1914
Offset: 0

Views

Author

Gus Wiseman, Feb 26 2020

Keywords

Comments

First differs from A000041 at a(6) = 10, A000041(6) = 11.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

			The a(1) = 1 through a(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (111111)  (2221)     (431)
                                               (1111111)  (521)
                                                          (2222)
                                                          (11111111)

Links

Crossrefs

The non-negated version is A332283.
The non-negated complement is counted by A332284.
The strict case is A332577.
The case of run-lengths (instead of differences) is A332638.
The complement is counted by A332744.
The Heinz numbers of partitions not in this class are A332287.
Unimodal compositions are A001523.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Cf. A007052, A332280, A332285, A332286, A332639, A332642, A332669, A332670, A332741.

Programs

  • Mathematica
    unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]

A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 28, 39, 55, 77, 107, 142, 194, 254, 332, 434, 563, 716, 919, 1162, 1464, 1841, 2305, 2857, 3555, 4383, 5394, 6617, 8099, 9859, 12006, 14551, 17600, 21236, 25574, 30688, 36809, 44007, 52527, 62574, 74430, 88304, 104675, 123799
Offset: 0

Views

Author

Gus Wiseman, Feb 27 2020

Keywords

Comments

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

Examples

			The a(4) = 1 through a(9) = 17 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (2211)   (511)     (611)      (711)
                 (3111)   (3211)    (3221)     (3222)
                 (21111)  (4111)    (3311)     (4221)
                          (22111)   (4211)     (4311)
                          (31111)   (5111)     (5211)
                          (211111)  (22211)    (6111)
                                    (32111)    (32211)
                                    (41111)    (33111)
                                    (221111)   (42111)
                                    (311111)   (51111)
                                    (2111111)  (222111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).

Links

Crossrefs

The complement is counted by A332728.
The non-negated version is A332284.
The strict case is A332579.
The case of run-lengths (instead of differences) is A332639.
The Heinz numbers of these partitions are A332832.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Numbers whose negated prime signature is not unimodal are A332642.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.

Programs

  • Mathematica
    unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],!unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]

A333191 Number of compositions of n whose run-lengths are either strictly increasing or strictly decreasing.

Original entry on oeis.org

1, 1, 2, 2, 5, 8, 10, 18, 24, 29, 44, 60, 68, 100, 130, 148, 201, 256, 310, 396, 478, 582, 736, 898, 1068, 1301, 1594, 1902, 2288, 2750, 3262, 3910, 4638, 5510, 6538, 7686, 9069, 10670, 12560, 14728, 17170, 20090, 23462, 27292, 31710, 36878, 42704, 49430
Offset: 0

Views

Author

Gus Wiseman, May 17 2020

Keywords

Comments

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

Examples

			The a(1) = 1 through a(7) = 18 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (211)   (221)    (222)     (223)
                    (1111)  (311)    (411)     (322)
                            (1112)   (1113)    (331)
                            (2111)   (3111)    (511)
                            (11111)  (11112)   (1114)
                                     (21111)   (1222)
                                     (111111)  (2221)
                                               (4111)
                                               (11113)
                                               (11122)
                                               (22111)
                                               (31111)
                                               (111112)
                                               (211111)
                                               (1111111)

Links

Crossrefs

The non-strict version is A332835.
The case of partitions is A333190.
Unimodal compositions are A001523.
Strict compositions are A032020.
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.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are unimodal or co-unimodal are A332746.
Compositions whose run-lengths are neither incr. nor decr. are A332833.
Compositions that are neither increasing nor decreasing are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are strictly incr. or strictly decr. are A333147.
Compositions with strictly increasing run-lengths are A333192.
Cf. A059204, A072706, A329398, A329744, A329766, A332641, A332726, A332831, A332745, A333149.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Less@@Length/@Split[#],Greater@@Length/@Split[#]]&]],{n,0,15}]

Formula

a(n > 0) = 2*A333192(n) - A000005(n).

Extensions

Terms a(26) and beyond from Giovanni Resta, May 19 2020

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}]

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

Views

Author

Gus Wiseman, May 17 2020

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Extensions

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

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

Views

Author

Gus Wiseman, Feb 29 2020

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
Also compositions whose run-lengths are not weakly decreasing.

Examples

			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).

Links

Crossrefs

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.

Programs

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

Formula

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

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020
Previous Showing 11-17 of 17 results.

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