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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).
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%I #18 Oct 24 2024 15:33:15

%S 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 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 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 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 From _Gus Wiseman_, May 21 2022: (Start)

%C 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 8 71 611 5111 41111 311111 2111111 11111111

%C 44 332 2222 22211 221111

%C 53 422 3221 32111

%C 62 431 3311

%C 521 4211

%C Indices of parts below the diagonal are also called strong nonexcedances.

%C (End)

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).

%D G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

%H John Tyler Rascoe, <a href="/A114088/b114088.txt">Rows n = 1..140, flattened</a>

%H Findstat, <a href="http://www.findstat.org/StatisticsDatabase/St000183/">St000183: The side length of the Durfee square of an integer partition</a>

%F G.f. = Sum_{k>=1} q^(k^2) / Product_{j=1..k} (1 - q^j)*(1 - t*q^j).

%F Sum_{k=0..n-1} k*T(n,k) = A114089(n).

%e 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 Triangle starts:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 2, 1, 1, 1;

%e 2, 2, 1, 1, 1;

%e 3, 3, 2, 1, 1, 1;

%e 3, 4, 3, 2, 1, 1, 1;

%p 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 subdiags[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];

%t Table[Length[Select[IntegerPartitions[n],subdiags[#]==k&]],{n,1,15},{k,0,n-1}] (* _Gus Wiseman_, May 21 2022 *)

%o (PARI)

%o 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 T_qt(10) \\ _John Tyler Rascoe_, Oct 24 2024

%Y Cf. A114087, A114089, A115995, A116365.

%Y Row sums: A000041.

%Y Column k = 0: A003114.

%Y Weak opposite: A115994.

%Y Permutations: A173018, weak A123125.

%Y Ordered: A352521, rank stat A352514, weak A352522.

%Y Opposite ordered: A352524, first col A008930, rank stat A352516.

%Y Weak opposite ordered: A352525, first col A177510, rank stat A352517.

%Y Weak: A353315.

%Y Opposite: A353318.

%Y A000700 counts self-conjugate partitions, ranked by A088902.

%Y A115720 counts partitions by Durfee square, rank stat A257990.

%Y A352490 gives the (strong) nonexcedance set of A122111, counted by A000701.

%Y Cf. A002620, A006918, A219282, A238352, A238874, A325039, A330644, A352487.

%K nonn,easy,tabl,changed

%O 1,7

%A _Emeric Deutsch_, Feb 12 2006