A139582 -id:A139582 - cp's OEIS frontend

A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

1, 1, 2, 1, 2, 3, 1, 3, 3, 5, 1, 3, 4, 5, 7, 1, 4, 5, 7, 7, 11, 1, 4, 6, 8, 9, 11, 15, 1, 5, 7, 11, 10, 15, 15, 22, 1, 5, 9, 12, 13, 17, 19, 22, 30, 1, 6, 10, 16, 15, 22, 21, 29, 30, 42, 1, 6, 12, 18, 19, 25, 26, 32, 38, 42, 56, 1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of "region" see A206437.

T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

			For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     3       4         5          7
.
Triangle begins:
1;
1,  2;
1,  2,  3;
1,  3,  3,  5;
1,  3,  4,  5,  7;
1,  4,  5,  7,  7, 11;
1,  4,  6,  8,  9, 11, 15;
1,  5,  7, 11, 10, 15, 15, 22;
1,  5,  9, 12, 13, 17, 19, 22, 30;
1,  6, 10, 16, 15, 22, 21, 29, 30, 42;
1,  6, 12, 18, 19, 25, 26, 32, 38, 42, 56;
1,  7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;

Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1.

Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446, A206437, A207779, A211978, A220517, A225598, A225600, A225610.

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

Original entry on oeis.org

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

Joshua Swanson, Nov 04 2020

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).

			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; ...

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

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

Cf. A000041, A139582.

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]

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

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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A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

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

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