A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 3, 4, 1, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 2, 0, 2, 4, 6, 3, 0

Offset

2,5

Comments

Row 1 is empty, so offset is 2.

Same as A386578 with rows reversed.

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Links

Table of n, a(n) for n=2..87.

Example

Row n = 21 counts the following permutations:
  .  111122  111221  111212  112121  .
     221111  112211  112112  121121
             122111  121112  121211
             211112  211121
                     211211
                     212111
Triangle begins:
  .
  1
  1  0
  0  2
  1  0  0
  0  2  1
  1  0  0  0
  0  0  6
  0  2  2  2
  0  2  2  0
  1  0  0  0  0
  0  0  6  6
  1  0  0  0  0  0
  0  2  3  0  0
  0  2  3  4  1
  0  0  0 24
  1  0  0  0  0  0  0
  0  0  6 12 12
  1  0  0  0  0  0  0  0
  0  0  6 12  2
  0  2  4  6  3  0

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
ugt[c_, x_]:=Select[Permutations[c], Function[q, Length[Select[Range[Length[q]-1], q[[#]]!=q[[#+1]]&]]==x]];
Table[Table[Length[ugt[nrmptn[n], k]], {k, 0, Length[nrmptn[n]]-1}], {n, 30}]

Crossrefs

Column k = 0 is A010051.

Row lengths are A056239.

Row sums are A318762.

Column k = last is A335125.

For prime indices we have A374252, reverse A386577.

Reversing all rows gives A386578.

A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.

A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.

A124762 gives inseparability of standard compositions, separability A333382.

A305936 is a multiset whose multiplicities are the prime indices of n.

A325534 counts separable multisets, ranks A335433, sums of A386583.

A325535 counts inseparable multisets, ranks A335448, sums of A386584.

A336106 counts partitions of separable type, ranks A335127, sums of A386585.

Cf. A001221, A001222, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A386581.

Sequence in context: A051127 A070176 A092606 * A374133 A275948 A356325

Adjacent sequences: A386576 A386577 A386578 * A386580 A386581 A386582

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 04 2025

Status

approved