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A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A246935 #27 Nov 12 2022 03:09:08
%S A246935 1,1,0,1,1,0,1,2,2,0,1,3,6,3,0,1,4,12,14,5,0,1,5,20,39,34,7,0,1,6,30,
%T A246935 84,129,74,11,0,1,7,42,155,356,399,166,15,0,1,8,56,258,805,1444,1245,
%U A246935 350,22,0,1,9,72,399,1590,4055,5876,3783,746,30,0
%N A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A246935 In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - _Vaclav Kotesovec_, Mar 19 2015
%C A246935 When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - _Geoffrey Critzer_, Nov 11 2022
%H A246935 Alois P. Heinz, <a href="/A246935/b246935.txt">Antidiagonals n = 0..140, flattened</a>
%H A246935 Kent E. Morrison, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F A246935 G.f. of column k: Product_{i>=1} 1/(1-k*x^i).
%F A246935 T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).
%e A246935 A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
%e A246935 Square array A(n,k) begins:
%e A246935   1,  1,   1,    1,     1,      1,      1,      1, ...
%e A246935   0,  1,   2,    3,     4,      5,      6,      7, ...
%e A246935   0,  2,   6,   12,    20,     30,     42,     56, ...
%e A246935   0,  3,  14,   39,    84,    155,    258,    399, ...
%e A246935   0,  5,  34,  129,   356,    805,   1590,   2849, ...
%e A246935   0,  7,  74,  399,  1444,   4055,   9582,  19999, ...
%e A246935   0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...
%e A246935   0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...
%p A246935 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246935       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A246935     end:
%p A246935 A:= (n, k)-> b(n$2, k):
%p A246935 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A246935 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; A[n_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 03 2015, after _Alois P. Heinz_ *)
%Y A246935 Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942.
%Y A246935 Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636.
%Y A246935 Main diagonal gives A124577.
%Y A246935 Cf. A144064, A255970, A256130.
%K A246935 nonn,tabl
%O A246935 0,8
%A A246935 _Alois P. Heinz_, Sep 08 2014