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A238394 Number of partitions of n that sorted in increasing order do not contain a part k in position k.

%I A238394 #29 Oct 27 2023 20:51:06
%S A238394 1,0,1,1,1,2,3,3,4,5,8,9,12,13,17,22,28,34,42,48,59,71,88,106,130,151,
%T A238394 181,210,250,295,354,417,494,577,675,780,909,1053,1231,1431,1668,1930,
%U A238394 2240,2573,2963,3392,3896,4461,5129,5873,6742,7710,8816,10043,11439
%N A238394 Number of partitions of n that sorted in increasing order do not contain a part k in position k.
%C A238394 The definition forbids partitions with a part equal to 1, so the smallest possible part is 2, which however can appear at most once.
%C A238394 Note that considering partitions in standard decreasing order, we obtain A064428.
%H A238394 Alois P. Heinz, <a href="/A238394/b238394.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from Giovanni Resta)
%F A238394 a(n) + A238395(n) = p(n) = A000041(n).
%F A238394 a(n) = Sum_{k>=0} A238406(n,k). - _Alois P. Heinz_, Feb 26 2014
%F A238394 a(n) = A238352(n,0). - _Alois P. Heinz_, Jun 08 2014
%e A238394 a(6) = 3, because of the 11 partitions of 6 only 3 do not contain a 1 in position 1, a 2 in position 2, or a 3 in position 3, namely (3,3), (2,4) and (6).
%e A238394 From _Joerg Arndt_, Mar 23 2014: (Start)
%e A238394 There are a(15) = 22 such partitions of 15:
%e A238394 01:  [ 2 3 4 6 ]
%e A238394 02:  [ 2 3 5 5 ]
%e A238394 03:  [ 2 3 10 ]
%e A238394 04:  [ 2 4 4 5 ]
%e A238394 05:  [ 2 4 9 ]
%e A238394 06:  [ 2 5 8 ]
%e A238394 07:  [ 2 6 7 ]
%e A238394 08:  [ 2 13 ]
%e A238394 09:  [ 3 3 4 5 ]
%e A238394 10:  [ 3 3 9 ]
%e A238394 11:  [ 3 4 8 ]
%e A238394 12:  [ 3 5 7 ]
%e A238394 13:  [ 3 6 6 ]
%e A238394 14:  [ 3 12 ]
%e A238394 15:  [ 4 4 7 ]
%e A238394 16:  [ 4 5 6 ]
%e A238394 17:  [ 4 11 ]
%e A238394 18:  [ 5 5 5 ]
%e A238394 19:  [ 5 10 ]
%e A238394 20:  [ 6 9 ]
%e A238394 21:  [ 7 8 ]
%e A238394 22:  [ 15 ]
%e A238394 (End)
%p A238394 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A238394       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
%p A238394        x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
%p A238394     end:
%p A238394 a:= n-> (p-> add(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A238394 seq(a(n), n=0..70);  # _Alois P. Heinz_, Feb 26 2014
%t A238394 a[n_] := Length@ Select[ IntegerPartitions@n, 0 < Min@ Abs[ Reverse@# - Range@ Length@#] &]; Array[a, 30]
%t A238394 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]]; a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]} ] ][b[n, n]]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Nov 02 2015, after _Alois P. Heinz_ *)
%Y A238394 Cf. A000041, A238395, A064428, A001522, A238352.
%K A238394 nonn
%O A238394 0,6
%A A238394 _Giovanni Resta_, Feb 26 2014