A325192 - OEIS

A325192

Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.

1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 2, 0, 0, 2, 1, 2, 2, 0, 0, 3, 2, 2, 2, 2, 0, 0, 2, 4, 3, 2, 2, 2, 0, 0, 1, 7, 4, 4, 2, 2, 2, 0, 1, 0, 6, 8, 5, 4, 2, 2, 2, 0, 0, 2, 5, 11, 8, 6, 4, 2, 2, 2, 0, 0, 3, 4, 12, 12, 9, 6, 4, 2, 2, 2, 0, 0, 4, 5, 13, 17, 12, 10, 6, 4, 2, 2, 2, 0

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OFFSET

0,5

COMMENTS

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

REFERENCES

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Wikipedia, Durfee square.

FORMULA

Sum_{k=1..n} k*T(n,k) = A368985(n) - A115995(n). - Andrew Howroyd, Jan 12 2024

EXAMPLE

Triangle begins:

1 0

0 2 0

0 1 2 0

1 0 2 2 0

0 2 1 2 2 0

0 3 2 2 2 2 0

0 2 4 3 2 2 2 0

0 1 7 4 4 2 2 2 0

1 0 6 8 5 4 2 2 2 0

0 2 5 11 8 6 4 2 2 2 0

0 3 4 12 12 9 6 4 2 2 2 0

0 4 5 13 17 12 10 6 4 2 2 2 0

0 3 9 12 20 18 13 10 6 4 2 2 2 0

0 2 12 15 23 25 18 14 10 6 4 2 2 2 0

0 1 15 19 26 30 26 19 14 10 6 4 2 2 2 0

Row 9 counts the following partitions (empty columns not shown):

333 432 54 63 72 711 81 9

441 522 621 6111 3111111 21111111 111111111

3222 531 51111 411111

3321 5211 222111 2211111

4221 22221 321111

4311 32211

33111

42111

MATHEMATICA

durf[ptn_]:=Length[Select[Range[Length[ptn]], ptn[[#]]>=#&]];

codurf[ptn_]:=Max[Length[ptn], Max[ptn]];

Table[Length[Select[IntegerPartitions[n], codurf[#]-durf[#]==k&]], {n, 0, 15}, {k, 0, n}]

PROG

(PARI) row(n)={my(r=vector(n+1)); if(n==0, r[1]=1, forpart(p=n, my(c=1); while(c<#p && c<p[#p-c], c++); r[max(#p, p[#p])-c+1]++)); r} \\ Andrew Howroyd, Jan 12 2024

CROSSREFS

Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182.

Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200.

Cf. A115995, A368985.

Sequence in context: A099584 A375159 A100563 * A087773 A025867 A078646

Adjacent sequences: A325189 A325190 A325191 * A325193 A325194 A325195

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Apr 08 2019

STATUS

approved