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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
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.
Sequence in context: A366888 A243924 A335474 * A272759 A272760 A054717
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2020
STATUS
approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)