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A338621
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Triangle read by rows: A(n, k) is the number of partitions of n with "aft" value k (see comments).
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1
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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
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OFFSET
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0,3
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COMMENTS
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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).
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REFERENCES
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S. C. Billey, M. Konvalinka, and J. P. Swanson, Asymptotic normality of the major index on standard tableaux, Adv. in Appl. Math. 113 (2020).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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; ...
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MATHEMATICA
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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]
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PROG
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(PARI) Row(n)={if(n==0, [1], my(v=vector(n)); forpart(p=n, v[1+n-max(#p, p[#p])]++); Vecrev(Polrev(v)))}
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CROSSREFS
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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STATUS
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approved
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