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 A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order. 9
 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, 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, 12, 11, 1, 2, 2, 4, 2, 5, 5, 7, 2, 4, 4, 11, 5, 14, 11, 12, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Number of ways to deal out the k-th composition in standard order to form a multiset of hands. 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. LINKS Table of n, a(n) for n=0..80. FORMULA For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A292884(n). EXAMPLE The dealings for n = 1, 3, 7, 11, 13, 23, 43: (1) (11) (111) (211) (121) (2111) (2211) (1)(1) (1)(11) (1)(21) (1)(12) (11)(21) (11)(22) (1)(1)(1) (2)(11) (1)(21) (1)(211) (1)(221) (1)(1)(2) (2)(11) (2)(111) (21)(21) (1)(1)(2) (1)(1)(21) (2)(211) (1)(2)(11) (1)(1)(22) (1)(1)(1)(2) (1)(2)(21) (2)(2)(11) (1)(1)(2)(2) MATHEMATICA nn=100; comps[0]:={{}}; comps[n_]:=Join@@Table[Prepend[#, i]&/@comps[n-i], {i, n}]; sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; dealings[q_]:=Union[Function[ptn, Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]]; Table[Length[dealings[stc[n]]], {n, 0, nn}] CROSSREFS Multisets of compositions are counted by A034691. Combinatory separations of normal multisets are counted by A269134. Dealings with total sum n are counted by A292884. Length of co-Lyndon factorization of binary expansion is A329312. Length of Lyndon factorization of reversed binary expansion is A329313. All of the following pertain to compositions in standard order (A066099): - Length is A000120. - Necklaces are A065609. - Sum is A070939. - Runs are counted by A124767. - Rotational symmetries are counted by A138904. - Strict compositions are A233564. - Constant compositions are A272919. - Lyndon words are A275692. - Co-Lyndon words are A326774. - Aperiodic compositions are A328594. - Length of Lyndon factorization is A329312. - Distinct rotations are counted by A333632. - Co-Lyndon factorizations are counted by A333765. - Lyndon factorizations are counted by A333940. - Length of co-Lyndon factorization is A334029. - Combinatory separations are A334030. Cf. A000031, A000740, A001037, A008965, A027375, A059966, A060223, A211100, A328595, A328596, A333764, A333943. Sequence in context: A366888 A243924 A335474 * A272759 A272760 A054717 Adjacent sequences: A333936 A333937 A333938 * A333940 A333941 A333942 KEYWORD nonn AUTHOR Gus Wiseman, Apr 15 2020 STATUS approved

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