A374703 Number of integer compositions of 2n whose leaders of weakly decreasing runs sum to n. Center n = 2*k of the triangle A374748.

1, 1, 2, 9, 24, 96, 343, 1242, 4700, 17352, 65995

Offset: 0

Author

Gus Wiseman, Aug 12 2024

Keywords

• nonn
• more

Comments

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

Examples

			The a(0) = 1 through a(4) = 24 compositions:
  ()  (11)  (22)   (33)     (44)
            (211)  (321)    (422)
                   (1122)   (431)
                   (1221)   (1133)
                   (3111)   (1322)
                   (11112)  (1331)
                   (11121)  (4211)
                   (11211)  (11132)
                   (12111)  (11321)
                            (13211)
                            (21122)
                            (21221)
                            (22112)
                            (22121)
                            (41111)
                            (111113)
                            (111131)
                            (111311)
                            (113111)
                            (131111)
                            (211112)
                            (211121)
                            (211211)
                            (212111)
		

Links

• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

For reversed partitions we have A364910.

For strictly decreasing runs we have the center of A374700.

Center n = 2*k of the triangle A374748.

A003242 counts anti-run compositions.

A011782 counts integer compositions.

A238130, A238279, A333755 count compositions by number of runs.

A274174 counts contiguous compositions, ranks A374249.

Cf. A000041, A004526, A106356, A124766, A188900, A238343, A261982, A333213.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[2n],Total[First/@Split[#,GreaterEqual]]==n&]],{n,0,8}]