

A333939


Number of multisets of compositions that can be shuffled together to obtain the kth 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
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OFFSET

0,4


COMMENTS

Number of ways to deal out the kth 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 kth 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^(n1)..2^n1} 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[ni], {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 coLyndon 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):
 Rotational symmetries are counted by A138904.
 Constant compositions are A272919.
 Aperiodic compositions are A328594.
 Length of Lyndon factorization is A329312.
 Distinct rotations are counted by A333632.
 CoLyndon factorizations are counted by A333765.
 Lyndon factorizations are counted by A333940.
 Length of coLyndon factorization is A334029.
 Combinatory separations are A334030.
Cf. A000031, A000740, A001037, A008965, A027375, A059966, A060223, A211100, A328595, A328596, A333764, A333943.


KEYWORD

nonn


AUTHOR



STATUS

approved



