cp's OEIS Frontend

A328891 Irregular table T(n,k) = #{m > 0: m occurs m times in the k-th partition of n, using A&S order (A036036)}, 1 <= k <= A000041(n), n >= 0.

%I A328891 #19 Jul 20 2020 18:57:02
%S A328891 0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,2,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,
%T A328891 0,1,1,1,0,0,1,0,1,0,0,0,1,0,0,0,0,1,1,1,0,0,0,0,2,0,0,0,0,0,1,0,0,0,
%U A328891 1,0,0,0,0,1,1,1,1,0,1,0,0,0,2,1,0,0,0,0,1,1,0,0,0,0,1,0,0,0,1
%N A328891 Irregular table T(n,k) = #{m > 0: m occurs m times in the k-th partition of n, using A&S order (A036036)}, 1 <= k <= A000041(n), n >= 0.
%C A328891 In the n-th row, the partitions of n are considered in the "Abramowitz and Stegun" or graded (reflected or not) colexicographic ordering, as in A036036 or A036037. For each partition this counts the numbers m > 0 such that there are exactly m parts equal to m in the partition.
%C A328891 Row lengths are A000041(n) = number of partitions of n, the partition numbers.
%H A328891 OEIS Wiki, <a href="/wiki/Orderings_of_partitions">Orderings of partitions</a>
%e A328891 The table reads:
%e A328891   n \ T(n,k), ...
%e A328891   0 : 0;   (The only partition of 0 is [], having no number at all in it.)
%e A328891   1 : 1;   (The only partition of 1 is [1], in which the number m=1 occurs 1 time.)
%e A328891   2 : 0,0;   (Neither [2] nor [1,1] have some m occurring m times.)
%e A328891   3 : 0,1,0;   ([3] and [1,1,1] have no m, but [1,2] has m=1 occurring m times.)
%e A328891   4 : 0,1,1,0,0;   (Here [1,3] and [2,2] have m=1 resp. m=2 occurring m times.)
%e A328891   5 : 0,1,0,0,2,0,0;   ([1,4] has m=1, [1,2,2] has m=1 and m=2 occurring m times.)
%e A328891   6 : 0,1,0,0,0,1,0,0,1,0,0;
%e A328891   7 : 0,1,0,0,0,1,1,1,0,0,1,0,1,0,0;
%e A328891   (...)
%e A328891 Column 1 = (0,1,0,...) = A063524, characteristic function of {1}: The corresponding partition is [n], except for [] when n=0.
%e A328891 Column 2 = (0,1,1,1,...) = signum(n-2) = A057427(n-2), n >= 2: The corresponding partition is [1, n-1].
%e A328891 Column 3 = A063524(n-3) = A185014(n), characteristic function of {4}: The corresponding partition is [2, n-2] for n >= 4, and [1,1,1] for n = 3.
%e A328891 Column 4 = (0,...) = A000004(n-4), the zero function: The corresponding partition is [3, n-3] for n >= 6, and [1,1,2] for n = 4 and [1,1,3] for n = 5.
%e A328891 Row sums = A276428(n) = sum over all partitions of n of the number of distinct parts m of multiplicity m.
%o A328891 (PARI) apply( A328891_row(n, r=[])={forpart(p=n, my(s, c=1); for(i=1, #p, p[i]==if(i<#p, p[i+1]) && c++ && next; c==p[i] && s++; c=1); r=concat(r,s));r}, [0..12])
%Y A328891 Cf. A036036 (list of partitions in Abramowitz & Stegun or graded reflected colexicographic order).
%Y A328891 Cf. A000041 (partition numbers = row lengths).
%Y A328891 Cf. A063524 (col.1: chi_{1}), A057427 (col.2: signum), A185014 (col.3: chi_{4}), A000004 (col.4: zero).
%Y A328891 Cf. A276427 (frequency of 0, ..., max.value in each row), A276428 (row sums), A276429, A276434, A277101.
%Y A328891 Cf. A328806 (row length of A276427(n) = 1 + largest value in row n).
%K A328891 nonn,tabf,easy
%O A328891 0,17
%A A328891 _M. F. Hasler_, Oct 29 2019