|
%I #12 Jan 12 2024 17:27:35
%S 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,
%T 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,
%U 4,12,12,9,6,4,2,2,2,0,0,4,5,13,17,12,10,6,4,2,2,2,0
%N 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.
%C 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.
%D Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.
%H Andrew Howroyd, <a href="/A325192/b325192.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Durfee_square">Durfee square</a>.
%F Sum_{k=1..n} k*T(n,k) = A368985(n) - A115995(n). - _Andrew Howroyd_, Jan 12 2024
%e Triangle begins:
%e 1
%e 1 0
%e 0 2 0
%e 0 1 2 0
%e 1 0 2 2 0
%e 0 2 1 2 2 0
%e 0 3 2 2 2 2 0
%e 0 2 4 3 2 2 2 0
%e 0 1 7 4 4 2 2 2 0
%e 1 0 6 8 5 4 2 2 2 0
%e 0 2 5 11 8 6 4 2 2 2 0
%e 0 3 4 12 12 9 6 4 2 2 2 0
%e 0 4 5 13 17 12 10 6 4 2 2 2 0
%e 0 3 9 12 20 18 13 10 6 4 2 2 2 0
%e 0 2 12 15 23 25 18 14 10 6 4 2 2 2 0
%e 0 1 15 19 26 30 26 19 14 10 6 4 2 2 2 0
%e Row 9 counts the following partitions (empty columns not shown):
%e 333 432 54 63 72 711 81 9
%e 441 522 621 6111 3111111 21111111 111111111
%e 3222 531 51111 411111
%e 3321 5211 222111 2211111
%e 4221 22221 321111
%e 4311 32211
%e 33111
%e 42111
%t durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
%t codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
%t Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==k&]],{n,0,15},{k,0,n}]
%o (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
%Y Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182.
%Y Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200.
%Y Cf. A115995, A368985.
%K nonn,tabl
%O 0,5
%A _Gus Wiseman_, Apr 08 2019
|
