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A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

%I A354234 #13 Jan 19 2023 22:36:27
%S A354234 1,2,1,3,1,1,5,3,1,1,7,4,2,1,1,11,7,4,2,1,1,15,10,6,3,2,1,1,22,16,9,6,
%T A354234 3,2,1,1,30,22,14,8,5,3,2,1,1,42,32,20,13,8,5,3,2,1,1,56,44,29,18,12,
%U A354234 7,5,3,2,1,1,77,62,41,27,17,12,7,5,3,2,1,1
%N A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.
%C A354234 Also partitions of n with at least one part appearing k or more times. It would be interesting to have a bijective proof of this.
%H A354234 Andrew Howroyd, <a href="/A354234/b354234.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e A354234 Triangle begins:
%e A354234    1
%e A354234    2  1
%e A354234    3  1  1
%e A354234    5  3  1  1
%e A354234    7  4  2  1  1
%e A354234   11  7  4  2  1  1
%e A354234   15 10  6  3  2  1  1
%e A354234   22 16  9  6  3  2  1  1
%e A354234   30 22 14  8  5  3  2  1  1
%e A354234   42 32 20 13  8  5  3  2  1  1
%e A354234   56 44 29 18 12  7  5  3  2  1  1
%e A354234   77 62 41 27 17 12  7  5  3  2  1  1
%e A354234 For example, row n = 5 counts the following partitions:
%e A354234   (5)      (32)    (32)   (41)  (5)
%e A354234   (32)     (41)    (311)
%e A354234   (41)     (221)
%e A354234   (221)    (2111)
%e A354234   (311)
%e A354234   (2111)
%e A354234   (11111)
%e A354234 At least one part appearing k or more times:
%e A354234   (5)      (221)    (2111)   (11111)  (11111)
%e A354234   (32)     (311)    (11111)
%e A354234   (41)     (2111)
%e A354234   (221)    (11111)
%e A354234   (311)
%e A354234   (2111)
%e A354234   (11111)
%t A354234 Table[Length[Select[IntegerPartitions[n],MemberQ[#/k,_?IntegerQ]&]],{n,1,15},{k,1,n}]
%t A354234 - or -
%t A354234 Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]>=k&]],{n,1,15},{k,1,n}]
%o A354234 (PARI) \\ here P(k,n) is partitions with no part divisible by k as g.f.
%o A354234 P(k,n)={1/prod(i=1, n, 1 - if(i%k, x^i) + O(x*x^n))}
%o A354234 M(n,m=n)={my(p=P(n+1,n)); Mat(vector(m, k, Col(p-P(k,n), -n) ))}
%o A354234 { my(A=M(12)); for(n=1, #A, print(A[n,1..n])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y A354234 The complement is counted by A061199.
%Y A354234 Differences of consecutive terms are A091602.
%Y A354234 Column k = 1 is A000041.
%Y A354234 Column k = 2 is A047967, ranked by A013929 and A324929.
%Y A354234 Column k = 3 is A295341, ranked by A046099 and A354235.
%Y A354234 Column k = 4 is A295342.
%Y A354234 A000041 counts integer partitions, strict A000009.
%Y A354234 A047966 counts uniform partitions.
%Y A354234 Cf. A002033, A006918, A064410, A117485, A238394, A238395, A325534.
%K A354234 nonn,tabl
%O A354234 1,2
%A A354234 _Gus Wiseman_, May 22 2022