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A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

%I A323522 #18 May 20 2021 04:08:41
%S A323522 1,1,1,1,1,1,1,1,1,1,25,25,49,73,121,145,217,265,361,433,553,649,817,
%T A323522 937,1129,1297,1537,1729,2017,2257,2593,2881,3265,3601,4057,4441,4945,
%U A323522 5401,5977,6481,7129,7705,8425,9073,9865,373465,374353,738025,1101865,1828513
%N A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.
%H A323522 Alois P. Heinz, <a href="/A323522/b323522.txt">Table of n, a(n) for n = 0..7000</a>
%F A323522 a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.
%e A323522 The a(10) = 25 matrices:
%e A323522   [10]
%e A323522 .
%e A323522   [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]
%e A323522   [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]
%e A323522 .
%e A323522   [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]
%e A323522   [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]
%e A323522 .
%e A323522   [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]
%e A323522   [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]
%p A323522 b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
%p A323522       -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
%p A323522     end:
%p A323522 a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):
%p A323522 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 17 2019
%t A323522 Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n,{k^2}],UnsameQ@@#&]],{k,n}],{n,20}]
%t A323522 (* Second program: *)
%t A323522 q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,
%t A323522      If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];
%t A323522 a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];
%t A323522 a /@ Range[0, 50] (* _Jean-François Alcover_, May 20 2021 *)
%Y A323522 Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529.
%K A323522 nonn
%O A323522 0,11
%A A323522 _Gus Wiseman_, Jan 17 2019