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Irregular triangle read by rows where if d|n then T(n,d) is the number of non-isomorphic multiset partitions of a multiset with d copies of each integer from 1 to n/d.
3

%I #11 Feb 02 2022 23:34:35

%S 1,2,2,3,3,5,7,5,7,7,11,23,21,11,15,15,22,79,66,22,30,162,30,42,274,

%T 192,42,56,56,77,1003,1636,1338,565,77,101,101,135,3763,1579,135,176,

%U 19977,10585,176,231,14723,43686,4348,231,297,297,385,59663,298416,82694,11582,385

%N Irregular triangle read by rows where if d|n then T(n,d) is the number of non-isomorphic multiset partitions of a multiset with d copies of each integer from 1 to n/d.

%H Andrew Howroyd, <a href="/A322787/b322787.txt">Table of n, a(n) for n = 1..207</a> (rows 1..50)

%e Triangle begins:

%e 1

%e 2 2

%e 3 3

%e 5 7 5

%e 7 7

%e 11 23 21 11

%e 15 15

%e 22 79 66 22

%e 30 162 30

%e 42 274 192 42

%e Non-isomorphic representatives of the multiset partitions counted under row 6:

%e {123456} {112233} {111222} {111111}

%e {1}{23456} {1}{12233} {1}{11222} {1}{11111}

%e {12}{3456} {11}{2233} {11}{1222} {11}{1111}

%e {123}{456} {112}{233} {111}{222} {111}{111}

%e {1}{2}{3456} {12}{1233} {112}{122} {1}{1}{1111}

%e {1}{23}{456} {123}{123} {12}{1122} {1}{11}{111}

%e {12}{34}{56} {1}{1}{2233} {1}{1}{1222} {11}{11}{11}

%e {1}{2}{3}{456} {1}{12}{233} {1}{11}{222} {1}{1}{1}{111}

%e {1}{2}{34}{56} {11}{22}{33} {11}{12}{22} {1}{1}{11}{11}

%e {1}{2}{3}{4}{56} {11}{23}{23} {1}{12}{122} {1}{1}{1}{1}{11}

%e {1}{2}{3}{4}{5}{6} {1}{2}{1233} {1}{2}{1122} {1}{1}{1}{1}{1}{1}

%e {12}{13}{23} {12}{12}{12}

%e {1}{23}{123} {2}{11}{122}

%e {2}{11}{233} {1}{1}{1}{222}

%e {1}{1}{2}{233} {1}{1}{12}{22}

%e {1}{1}{22}{33} {1}{1}{2}{122}

%e {1}{1}{23}{23} {1}{2}{11}{22}

%e {1}{2}{12}{33} {1}{2}{12}{12}

%e {1}{2}{13}{23} {1}{1}{1}{2}{22}

%e {1}{2}{3}{123} {1}{1}{2}{2}{12}

%e {1}{1}{2}{2}{33} {1}{1}{1}{2}{2}{2}

%e {1}{1}{2}{3}{23}

%e {1}{1}{2}{2}{3}{3}

%o (PARI) \\ See A318951 for RowSumMats

%o row(n)={my(d=divisors(n)); vector(#d, i, RowSumMats(n/d[i], n, d[i]))}

%o { for(n=1, 15, print(row(n))) } \\ _Andrew Howroyd_, Feb 02 2022

%Y Row sums are A306017. First column is A000041.

%Y Cf. A001055, A005176, A056239, A072774, A100778, A295193, A306018, A318951, A319190, A319612, A322784, A322785, A322788, A322792.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Dec 26 2018

%E Terms a(28) and beyond from _Andrew Howroyd_, Feb 02 2022