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.

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, 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, 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

Offset: 0

Author

M. F. Hasler, Oct 29 2019

Keywords

• nonn
• tabf
• easy

Comments

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.

Row lengths are A000041(n) = number of partitions of n, the partition numbers.

Examples

			The table reads:
  n \ T(n,k), ...
  0 : 0;   (The only partition of 0 is [], having no number at all in it.)
  1 : 1;   (The only partition of 1 is [1], in which the number m=1 occurs 1 time.)
  2 : 0,0;   (Neither [2] nor [1,1] have some m occurring m times.)
  3 : 0,1,0;   ([3] and [1,1,1] have no m, but [1,2] has m=1 occurring m times.)
  4 : 0,1,1,0,0;   (Here [1,3] and [2,2] have m=1 resp. m=2 occurring m times.)
  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.)
  6 : 0,1,0,0,0,1,0,0,1,0,0;
  7 : 0,1,0,0,0,1,1,1,0,0,1,0,1,0,0;
  (...)
Column 1 = (0,1,0,...) = A063524, characteristic function of {1}: The corresponding partition is [n], except for [] when n=0.
Column 2 = (0,1,1,1,...) = signum(n-2) = A057427(n-2), n >= 2: The corresponding partition is [1, n-1].
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.
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.
Row sums = A276428(n) = sum over all partitions of n of the number of distinct parts m of multiplicity m.
		

Links

• OEIS Wiki, Orderings of partitions

Crossrefs

Cf. A036036 (list of partitions in Abramowitz & Stegun or graded reflected colexicographic order).

Cf. A000041 (partition numbers = row lengths).

Cf. A063524 (col.1: chi_{1}), A057427 (col.2: signum), A185014 (col.3: chi_{4}), A000004 (col.4: zero).

Cf. A276427 (frequency of 0, ..., max.value in each row), A276428 (row sums), A276429, A276434, A277101.

Cf. A328806 (row length of A276427(n) = 1 + largest value in row n).

Programs

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])