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

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

0,3

Comments

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

Links

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

Example

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).

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 *)

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.

Sequence in context: A362610 A363260 A195012 * A323092 A238594 A350837

Adjacent sequences: A333189 A333190 A333191 * A333193 A333194 A333195

Keyword

nonn

Author

Gus Wiseman, May 17 2020

Extensions

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

Status

approved