A338621 - OEIS

%I #25 Dec 23 2020 07:36:56

%S 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,

%T 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,

%U 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

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

%C 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).

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

%C Row sums are partition numbers A000041.

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

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

%H S. C. Billey, M. Konvalinka, and J. P. Swanson, <a href="https://arxiv.org/abs/1905.00975">Asymptotic normality of the major index on standard tableaux</a>, arXiv:1905.00975 [math.CO], 2019.

%H FindStat - Combinatorial Statistic Finder, <a href="https://www.findstat.org/StatisticsDatabase/St001214/">The aft of an integer partition</a>

%F 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).

%e 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).

%e Triangle starts:

%e   1;

%e   1;

%e   2;

%e   2, 1;

%e   2, 2, 1;

%e   2, 2, 3;

%e   2, 2, 4, 3;

%e   2, 2, 4, 5,  2;

%e   2, 2, 4, 6,  7,  1;

%e   2, 2, 4, 6,  9,  6,  1;

%e   2, 2, 4, 6, 10, 11,  7;

%e   2, 2, 4, 6, 10, 13, 14,  5;

%e   2, 2, 4, 6, 10, 14, 19, 15,  5;

%e   2, 2, 4, 6, 10, 14, 21, 22, 17,  3;

%e   2, 2, 4, 6, 10, 14, 22, 27, 29, 17,  2;

%e   2, 2, 4, 6, 10, 14, 22, 29, 36, 33, 17,  1;

%e   2, 2, 4, 6, 10, 14, 22, 30, 41, 45, 39, 15,  1;

%e   2, 2, 4, 6, 10, 14, 22, 30, 43, 52, 57, 41, 14;

%e   2, 2, 4, 6, 10, 14, 22, 30, 44, 57, 69, 67, 47, 11;

%e   2, 2, 4, 6, 10, 14, 22, 30, 44, 59, 76, 85, 81, 46, 9; ...

%t CoefficientList[

%t SeriesCoefficient[

%t   1 + Sum[If[r == 0, 1, 2] q^(r + 1) Sum[

%t       q^(2 s) t^s QBinomial[2 s + r, s, q t], {s, 0, 30}], {r, 0,

%t      30}], {q, 0, 20}], t]

%o (PARI) Row(n)={if(n==0, [1], my(v=vector(n)); forpart(p=n, v[1+n-max(#p, p[#p])]++); Vecrev(Polrev(v)))}

%o { for(n=1, 15, print(Row(n))) } \\ _Andrew Howroyd_, Nov 04 2020

%Y Cf. A000041, A139582.

%K easy,nonn,tabf

%O 0,3

%A _Joshua Swanson_, Nov 04 2020