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

0,3

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).

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

Table of n, a(n) for n=0..88.

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

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).

Example

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; ...

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]

Prog

(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

Crossrefs

Cf. A000041, A139582.

Sequence in context: A129706 A160384 A178305 * A024327 A073044 A361870

Adjacent sequences: A338618 A338619 A338620 * A338622 A338623 A338624

Keyword

easy,nonn,tabf

Author

Joshua Swanson, Nov 04 2020

Status

approved