%I #6 Apr 16 2020 18:48:33

%S 1,1,1,2,1,2,2,3,1,2,2,4,2,5,4,5,1,2,2,4,2,4,5,7,2,5,4,10,4,10,7,7,1,

%T 2,2,4,2,5,5,7,2,5,3,9,5,13,11,12,2,5,5,10,5,11,13,18,4,10,9,20,7,18,

%U 12,11,1,2,2,4,2,5,5,7,2,4,4,11,5,14,11,12,2

%N Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

%C Number of ways to deal out the k-th composition in standard order to form a multiset of hands.

%C A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%F For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A292884(n).

%e The dealings for n = 1, 3, 7, 11, 13, 23, 43:

%e (1) (11) (111) (211) (121) (2111) (2211)

%e (1)(1) (1)(11) (1)(21) (1)(12) (11)(21) (11)(22)

%e (1)(1)(1) (2)(11) (1)(21) (1)(211) (1)(221)

%e (1)(1)(2) (2)(11) (2)(111) (21)(21)

%e (1)(1)(2) (1)(1)(21) (2)(211)

%e (1)(2)(11) (1)(1)(22)

%e (1)(1)(1)(2) (1)(2)(21)

%e (2)(2)(11)

%e (1)(1)(2)(2)

%t nn=100;

%t comps[0]:={{}};comps[n_]:=Join@@Table[Prepend[#,i]&/@comps[n-i],{i,n}];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t dealings[q_]:=Union[Function[ptn,Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];

%t Table[Length[dealings[stc[n]]],{n,0,nn}]

%Y Multisets of compositions are counted by A034691.

%Y Combinatory separations of normal multisets are counted by A269134.

%Y Dealings with total sum n are counted by A292884.

%Y Length of co-Lyndon factorization of binary expansion is A329312.

%Y Length of Lyndon factorization of reversed binary expansion is A329313.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Necklaces are A065609.

%Y - Sum is A070939.

%Y - Runs are counted by A124767.

%Y - Rotational symmetries are counted by A138904.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Lyndon words are A275692.

%Y - Co-Lyndon words are A326774.

%Y - Aperiodic compositions are A328594.

%Y - Length of Lyndon factorization is A329312.

%Y - Distinct rotations are counted by A333632.

%Y - Co-Lyndon factorizations are counted by A333765.

%Y - Lyndon factorizations are counted by A333940.

%Y - Length of co-Lyndon factorization is A334029.

%Y - Combinatory separations are A334030.

%Y Cf. A000031, A000740, A001037, A008965, A027375, A059966, A060223, A211100, A328595, A328596, A333764, A333943.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 15 2020