cp's OEIS Frontend

A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.

%I A238875 #33 Apr 14 2025 09:02:23
%S A238875 1,1,1,2,2,4,5,7,10,15,18,26,35,47,61,80,103,138,175,224,283,362,455,
%T A238875 577,721,898,1111,1380,1701,2106,2577,3156,3844,4680,5671,6879,8312,
%U A238875 10034,12060,14478,17319,20715,24703,29442,35004,41578,49247,58278,68796,81132,95502,112320,131877,154705,181158,211908,247475
%N A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.
%C A238875 The partitions are represented as weakly increasing lists of parts.
%C A238875 Partitions with subdiagonal growth (A238876) with first part = 1.
%H A238875 Andrew Howroyd, <a href="/A238875/b238875.txt">Table of n, a(n) for n = 0..1000</a>
%H A238875 M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, <a href="https://doi.org/10.1007/s13370-025-01282-0">Subdiagonal and superdiagonal partitions</a>, Afr. Mat. 36, 77 (2025). See p. 9.
%F A238875 G.f.: Sum_{n>=0} x^n * P(n) where P(n) is the row polynomial of the n-th row of A129176. This works because A129176(j,k) is also the number of subdiagonal partitions of j+k with j parts. - _John Tyler Rascoe_, Jan 20 2024
%e A238875 The a(11) = 26 such partitions of 11 are:
%e A238875   01:  [ 1 1 1 1 1 1 1 1 1 1 1 ]
%e A238875   02:  [ 1 1 1 1 1 1 1 1 1 2 ]
%e A238875   03:  [ 1 1 1 1 1 1 1 1 3 ]
%e A238875   04:  [ 1 1 1 1 1 1 1 2 2 ]
%e A238875   05:  [ 1 1 1 1 1 1 1 4 ]
%e A238875   06:  [ 1 1 1 1 1 1 2 3 ]
%e A238875   07:  [ 1 1 1 1 1 1 5 ]
%e A238875   08:  [ 1 1 1 1 1 2 2 2 ]
%e A238875   09:  [ 1 1 1 1 1 2 4 ]
%e A238875   10:  [ 1 1 1 1 1 3 3 ]
%e A238875   11:  [ 1 1 1 1 1 6 ]
%e A238875   12:  [ 1 1 1 1 2 2 3 ]
%e A238875   13:  [ 1 1 1 1 2 5 ]
%e A238875   14:  [ 1 1 1 1 3 4 ]
%e A238875   15:  [ 1 1 1 2 2 2 2 ]
%e A238875   16:  [ 1 1 1 2 2 4 ]
%e A238875   17:  [ 1 1 1 2 3 3 ]
%e A238875   18:  [ 1 1 1 3 5 ]
%e A238875   19:  [ 1 1 1 4 4 ]
%e A238875   20:  [ 1 1 2 2 2 3 ]
%e A238875   21:  [ 1 1 2 2 5 ]
%e A238875   22:  [ 1 1 2 3 4 ]
%e A238875   23:  [ 1 1 3 3 3 ]
%e A238875   24:  [ 1 2 2 2 2 2 ]
%e A238875   25:  [ 1 2 2 2 4 ]
%e A238875   26:  [ 1 2 2 3 3 ]
%o A238875 (PARI) \\ here b: nr parts; k: max part, b+w-1: partition sum.
%o A238875 seq(n)={my(M=matrix(n,1), v=vector(n+1)); M[1,1]=v[1]=v[2]=1; for(b=2, n, M=matrix(n-b+1,b,w,k, if(w>=k, sum(j=1, min(b-1,k), M[w+1-k,j]))); v+=concat(vector(b),vecsum(Vec(M))~)); v} \\ _Andrew Howroyd_, Jan 19 2024
%o A238875 (PARI)
%o A238875 N=55;
%o A238875 VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
%o A238875 P(n) = VP[n+1];
%o A238875 for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
%o A238875 x='x+O('x^N);
%o A238875 A(x) = sum(n=0, N, x^n * P(n) );
%o A238875 Vec(A(x)) \\ _Joerg Arndt_, Jan 23 2024
%Y A238875 Cf. A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
%Y A238875 Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
%Y A238875 Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
%Y A238875 Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
%Y A238875 Cf. A129176 and A227543.
%K A238875 nonn
%O A238875 0,4
%A A238875 _Joerg Arndt_, Mar 24 2014