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A176723 Characteristic array for partitions which define multiset repetition classes. 12
1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0

COMMENTS

For the definition of multisets see, e.g., Stanley, p. 15.

Partitions of natural numbers n are ordered according to Abramowitz-Stegun (A-St) order.

Partitions of n>=1 are written in the exponent form (1^e[1], 2^e[2], 3^e[3],..., n^e[n]) with e[j] nonnegative numbers, for j=1,2,...,n, sum(e[j],j=1..n)=m (number of parts), and sum(j*e[j],j=1..n)=n. The empty partition for n=0 defines the empty multiset. In A115621 the multiset/partition of positive exponents is called the signature of the partition.

The classes being represented are the classes with the same signature.

Definition of multiset repetition class defining partitions: Every m part partition of n which has positive nonincreasing exponents defines a representative of a multiset repetition class of order m (a special m-multiset); i.e., the exponents of such

partitions satisfy e[1]>= e[2]>= ... >= e[M]>= 1 with largest part M. This will satisfy T(M) <= n where T(M) = A000217(M) is the sequence of triangular numbers; for n>=1 every sufficiently small positive M does occur.

Note that for each multiset repetition class the chosen defining partition (its representative) is the one with least n.

See below for some examples.

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

R. P. Stanley, Enumerative Combinatorics, Cambridge University Press, Vol. 1, 1999.

LINKS

Table of n, a(n) for n=0..105.

Wolfdieter Lang, First 20 rows, row sums and relevant partitions for n=1..15.

FORMULA

a(n,k)=1 if the k-th partition of n in A-St order (see above for the abbreviation A-St and a reference) is multiset defining and 0 else. The definition of a multiset repetition class defining partition is given above. See the examples below.

EXAMPLE

[1]; [1]; [0|1]; [0|1|1]; [0|0,0|1|1]; [0|0,0|0,0|1|1]; [0|0,0,0|0,1,0|0,1|1|1];... For each row n (separated by ;) the | separates partitions with different number of parts.

For n=6 the entry 1 at the 6th position stands for the partition (1^1,2^1,3^1)=(1,2,3) in A-St order.

The m=3 multiset corresponding to partition (1,2,3) coincides with the ordinary 3-set {1,2,3}.

Partition (1^4,2^1) = (1^4,2) (marking the next to last entry in row n=6) corresponds to the (m=5)-multiset {1,1,1,1,2}.

CROSSREFS

Row lengths A000041, row sums A007294, corresponding triangle with like m positions summed A176724.

Cf. A012257, A036036, A115621.

Sequence in context: A267015 A189292 A254634 * A189126 A189163 A189129

Adjacent sequences:  A176720 A176721 A176722 * A176724 A176725 A176726

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Jul 14 2010

EXTENSIONS

Edited by the author and Franklin T. Adams-Watters, Apr 02 2011

STATUS

approved

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Last modified March 25 09:40 EDT 2017. Contains 284060 sequences.