A338621 Triangle read by rows: A(n, k) is the number of partitions of n with "aft" value k (see comments).

1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 4, 5, 2, 2, 2, 4, 6, 7, 1, 2, 2, 4, 6, 9, 6, 1, 2, 2, 4, 6, 10, 11, 7, 2, 2, 4, 6, 10, 13, 14, 5, 2, 2, 4, 6, 10, 14, 19, 15, 5, 2, 2, 4, 6, 10, 14, 21, 22, 17, 3, 2, 2, 4, 6, 10, 14, 22, 27, 29, 17, 2, 2, 2, 4, 6, 10, 14, 22, 29, 36, 33, 17

Offset: 0

Author

Joshua Swanson, Nov 04 2020

Keywords

• easy
• nonn
• tabf

Comments

The "aft" of an integer partition is the number of cells minus the larger of the number of parts or the largest part. For example, aft(4, 2, 2) = 8-4 = 4 = aft(3, 3, 1, 1).

Columns stabilize to twice the partition numbers: A(n, k) = 2p(n) = A139582(n) if n > 2k.

Row sums are partition numbers A000041.

Maximum value of k in row n is n - ceiling(sqrt(n)) = (n-1) - floor(sqrt(n-1)) = A028391(n-1).

Examples

			A(6, 2) = 4 since there are four partitions with 6 cells and aft 2, namely (4, 2), (2, 2, 1, 1), (4, 1, 1), (3, 1, 1, 1).
Triangle starts:
  1;
  1;
  2;
  2, 1;
  2, 2, 1;
  2, 2, 3;
  2, 2, 4, 3;
  2, 2, 4, 5,  2;
  2, 2, 4, 6,  7,  1;
  2, 2, 4, 6,  9,  6,  1;
  2, 2, 4, 6, 10, 11,  7;
  2, 2, 4, 6, 10, 13, 14,  5;
  2, 2, 4, 6, 10, 14, 19, 15,  5;
  2, 2, 4, 6, 10, 14, 21, 22, 17,  3;
  2, 2, 4, 6, 10, 14, 22, 27, 29, 17,  2;
  2, 2, 4, 6, 10, 14, 22, 29, 36, 33, 17,  1;
  2, 2, 4, 6, 10, 14, 22, 30, 41, 45, 39, 15,  1;
  2, 2, 4, 6, 10, 14, 22, 30, 43, 52, 57, 41, 14;
  2, 2, 4, 6, 10, 14, 22, 30, 44, 57, 69, 67, 47, 11;
  2, 2, 4, 6, 10, 14, 22, 30, 44, 59, 76, 85, 81, 46, 9; ...
		

References

• S. C. Billey, M. Konvalinka, and J. P. Swanson, Asymptotic normality of the major index on standard tableaux, Adv. in Appl. Math. 113 (2020).

Links

• S. C. Billey, M. Konvalinka, and J. P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.
• FindStat - Combinatorial Statistic Finder, The aft of an integer partition

Crossrefs

Cf. A000041, A139582.

Programs

Mathematica

CoefficientList[
SeriesCoefficient[
  1 + Sum[If[r == 0, 1, 2] q^(r + 1) Sum[
      q^(2 s) t^s QBinomial[2 s + r, s, q t], {s, 0, 30}], {r, 0,
     30}], {q, 0, 20}], t]

PARI

Row(n)={if(n==0, [1], my(v=vector(n)); forpart(p=n, v[1+n-max(#p, p[#p])]++); Vecrev(Polrev(v)))}
{ for(n=1, 15, print(Row(n))) } \\ Andrew Howroyd, Nov 04 2020

Formula

G.f.: Sum_{lambda} t^aft(lambda) * q^|lambda| = 1 + Sum_{r >= 0} c_r * q^(r+1) * Sum_{s >= 0} q^(2*s) * t^s * [2*s + r, s]_(q*t) where c_0 = 1, c_r = 2 for r >= 1, and [a, b]_q is a Gaussian binomial coefficient (see A022166).