A176723 - OEIS

%I #24 Jun 09 2018 13:41:10

%S 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,

%T 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,

%U 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

%N Characteristic array for partitions which define multiset repetition classes.

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

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

%C 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.

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

%C 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.

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

%C See below for some examples.

%D 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.

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

%H Wolfdieter Lang, <a href="/A176723/a176723.pdf">First 20 rows, row sums and relevant partitions for n=1..15.</a>

%F 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.

%e [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.

%e 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.

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

%e 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}.

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

%Y Cf. A012257, A036036, A115621.

%K nonn,easy,tabf

%O 0

%A _Wolfdieter Lang_, Jul 14 2010

%E Edited by the author and _Franklin T. Adams-Watters_, Apr 02 2011