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
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
KEYWORD
easy,nonn,tabf
AUTHOR
Joshua Swanson, Nov 04 2020
STATUS
approved