cp's OEIS Frontend

A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

%I A325191 #23 Feb 16 2025 08:33:58
%S A325191 0,0,2,0,3,3,0,4,6,4,0,5,10,10,5,0,6,15,20,15,6,0,7,21,35,35,21,7,0,8,
%T A325191 28,56,70,56,28,8,0,9,36,84,126,126,84,36,9,0,10,45,120,210,252,210,
%U A325191 120,45,10,0,11,55,165,330,462
%N A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.
%C A325191 The Heinz numbers of these partitions are given by A325196.
%C A325191 Under the Bulgarian solitaire step, these partitions form cycles of length >= 2.  Length >= 2 means not the length=1 self-loop which occurs from the triangular partition when n is a triangular number.  See A074909 for self-loops included. - _Kevin Ryde_, Sep 27 2019
%H A325191 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000380">St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition</a>
%H A325191 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000384">St000384: The maximal part of the shifted composition of an integer partition</a>
%H A325191 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000783">St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram</a>
%H A325191 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>
%F A325191 Positions of zeros are A000217 = n * (n + 1) / 2.
%F A325191 a(n) = A074909(n) - A010054(n). - _Kevin Ryde_, Sep 27 2019
%e A325191 The a(2) = 2 through a(12) = 10 partitions (empty columns not shown):
%e A325191   (2)   (22)   (32)   (322)   (332)   (432)   (4322)   (4332)
%e A325191   (11)  (31)   (221)  (331)   (422)   (3321)  (4331)   (4422)
%e A325191         (211)  (311)  (421)   (431)   (4221)  (4421)   (4431)
%e A325191                       (3211)  (3221)  (4311)  (5321)   (5322)
%e A325191                               (3311)          (43211)  (5331)
%e A325191                               (4211)                   (5421)
%e A325191                                                        (43221)
%e A325191                                                        (43311)
%e A325191                                                        (44211)
%e A325191                                                        (53211)
%t A325191 otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325191 otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325191 Table[Length[Select[IntegerPartitions[n],otb[#]+1==otbmax[#]&]],{n,0,30}]
%o A325191 (PARI) a(n) = my(t=ceil(sqrtint(8*n+1)/2), r=n-t*(t-1)/2); if(r==0,0, binomial(t,r)); \\ _Kevin Ryde_, Sep 27 2019
%Y A325191 Column k=1 of A325200.
%Y A325191 Cf. A060687, A065770, A071724, A256617, A325166, A325169, A325178, A325179, A325181, A325187, A325188, A325189, A325195, A325196.
%K A325191 nonn,look
%O A325191 0,3
%A A325191 _Gus Wiseman_, Apr 11 2019