cp's OEIS Frontend

A114088 Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).

%I A114088 #18 Oct 24 2024 15:33:15
%S A114088 1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,3,3,2,1,1,1,3,4,3,2,1,1,1,4,5,5,3,2,1,
%T A114088 1,1,5,6,6,5,3,2,1,1,1,6,8,8,7,5,3,2,1,1,1,7,10,10,9,7,5,3,2,1,1,1,9,
%U A114088 13,13,12,10,7,5,3,2,1,1,1,10,16,17,15,13,10,7,5,3,2,1,1,1,12,20,22,20,17
%N A114088 Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).
%C A114088 From _Gus Wiseman_, May 21 2022: (Start)
%C A114088 Also the number of integer partitions of n with k parts below the diagonal. For example, the partition (3,2,2,1) has two parts (at positions 3 and 4) below the diagonal (1,2,3,4). Row n = 8 counts the following partitions:
%C A114088   8    71    611    5111    41111    311111   2111111   11111111
%C A114088   44   332   2222   22211   221111
%C A114088   53   422   3221   32111
%C A114088   62   431   3311
%C A114088        521   4211
%C A114088 Indices of parts below the diagonal are also called strong nonexcedances.
%C A114088 (End)
%D A114088 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
%D A114088 G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
%H A114088 John Tyler Rascoe, <a href="/A114088/b114088.txt">Rows n = 1..140, flattened</a>
%H A114088 Findstat, <a href="http://www.findstat.org/StatisticsDatabase/St000183/">St000183: The side length of the Durfee square of an integer partition</a>
%F A114088 G.f. = Sum_{k>=1} q^(k^2) / Product_{j=1..k} (1 - q^j)*(1 - t*q^j).
%F A114088 Sum_{k=0..n-1} k*T(n,k) = A114089(n).
%e A114088 T(7,2)=3 because we have [5,1,1], [3,2,1,1] and [2,2,2,1] (the bottom tails are [1,1], [1,1] and [2,1], respectively).
%e A114088 Triangle starts:
%e A114088   1;
%e A114088   1, 1;
%e A114088   1, 1, 1;
%e A114088   2, 1, 1, 1;
%e A114088   2, 2, 1, 1, 1;
%e A114088   3, 3, 2, 1, 1, 1;
%e A114088   3, 4, 3, 2, 1, 1, 1;
%p A114088 g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j),j=1..k),k=1..20): gserz:=simplify(series(g,z=0,30)): for n from 1 to 14 do P[n]:=coeff(gserz,z^n) od: for n from 1 to 14 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form
%t A114088 subdiags[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
%t A114088 Table[Length[Select[IntegerPartitions[n],subdiags[#]==k&]],{n,1,15},{k,0,n-1}] (* _Gus Wiseman_, May 21 2022 *)
%o A114088 (PARI)
%o A114088 T_qt(max_row) = {my(N=max_row+1, q='q+O('q^N), h = sum(k=1,N, q^(k^2)/prod(j=1,k, (1-q^j)*(1-t*q^j))) ); for(i=1, N-1, print(Vecrev(polcoef(h, i))))}
%o A114088 T_qt(10) \\ _John Tyler Rascoe_, Oct 24 2024
%Y A114088 Cf. A114087, A114089, A115995, A116365.
%Y A114088 Row sums: A000041.
%Y A114088 Column k = 0: A003114.
%Y A114088 Weak opposite: A115994.
%Y A114088 Permutations: A173018, weak A123125.
%Y A114088 Ordered: A352521, rank stat A352514, weak A352522.
%Y A114088 Opposite ordered: A352524, first col A008930, rank stat A352516.
%Y A114088 Weak opposite ordered: A352525, first col A177510, rank stat A352517.
%Y A114088 Weak: A353315.
%Y A114088 Opposite: A353318.
%Y A114088 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A114088 A115720 counts partitions by Durfee square, rank stat A257990.
%Y A114088 A352490 gives the (strong) nonexcedance set of A122111, counted by A000701.
%Y A114088 Cf. A002620, A006918, A219282, A238352, A238874, A325039, A330644, A352487.
%K A114088 nonn,easy,tabl
%O A114088 1,7
%A A114088 _Emeric Deutsch_, Feb 12 2006