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A136624 Irregular triangle read by rows: classify each numeric partition by sum of its parts and by the size of the staircase Ferrers board required to contain it. The triangle gives the number of partitions in each class, cf. A136102 and A136103.

%I A136624 #13 Feb 26 2025 11:38:17
%S A136624 1,1,2,1,2,3,3,1,2,2,6,7,6,4,1,2,2,4,8,12,15,17,14,10,5,1,2,2,4,6,12,
%T A136624 15,23,30,39,42,40,35,25,15,6,1,2,2,4,6,10,16,23,29,42,56,71,88,103,
%U A136624 112,114,102,86,65,41,21,7,1,2,2,4,6,10,14,24,31,43
%N A136624 Irregular triangle read by rows: classify each numeric partition by sum of its parts and by the size of the staircase Ferrers board required to contain it. The triangle gives the number of partitions in each class, cf. A136102 and A136103.
%C A136624 Sequences A136102 and A136103 encode the numeric partitions by least prime signature and the Ferrers boards by 1 2 12 360 75600 174636000 ... A006939.
%H A136624 John Tyler Rascoe, <a href="/A136624/b136624.txt">Rows n = 0..16, flattened</a>
%e A136624 Starting a new row each time we are required to use a larger Ferrer board the triangle begins:
%e A136624   1
%e A136624   ..1
%e A136624   .....2...1
%e A136624   .........2...3...3...1
%e A136624   .............2...2...6...7...6...4...1
%e A136624   .................2...2...4...8..12..15..17..14..10...5...1
%e A136624   .....................2...2...4...6..12..15..23..30..39..42..40..35..25..15..6..1
%o A136624 (PARI)
%o A136624 d(s,n) = {my(v = setminus([1..n],s), r=[], c=1); for(i=2, #v, if(v[i]==v[i-1]+1, c++ , r=concat(r, c); c=1)) ; return(concat(r, c))}
%o A136624 tri(n) = {n*(n+1)/2}
%o A136624 S(n) = {my(R = x^tri(n)); if(n<1, return(1), for(i=1,n-1, forsubset([n,i], s, my(u=d(vecextract([1..n],s),n)); R+=(x^(tri(n)-sum(j=1,#u, tri(u[j]))))*prod(j=1,#u, sum(z=0,u[j]-1, S(z))))); return(R))}
%o A136624 A136624(row_n) = {Vecrev(S(row_n)/x^(row_n))} \\ _John Tyler Rascoe_, Feb 25 2025
%Y A136624 Cf. A000041 (column sums), A000108, A006939, A025487, A071724 (row sums), A136102, A136103, A136625.
%K A136624 nonn,tabf
%O A136624 0,3
%A A136624 _Alford Arnold_, Jan 17 2008
%E A136624 a(26) onwards from _John Tyler Rascoe_, Feb 25 2025