A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

1, 1, 0, 1, 0, 1, 10, 4, 14, 84, 1136, 967, 3342, 12823, 101762, 1769580, 2492516, 8760576, 39134672, 219118191, 2159119722, 49449831729, 94989216228, 405379324204, 1767935161322, 11445339830314, 79463315035786, 1030079501685941, 29833791645100204

Offset

0,7

Comments

An anti-run is a sequence with no adjacent equal terms.

Links

Christian Sievers, Table of n, a(n) for n = 0..100

Example

The a(7) = 4 anti-runs are:
  (1,2,1,2,1,2,1)
  (1,2,1,2,1,3,1)
  (1,2,1,3,1,2,1)
  (1,3,1,2,1,2,1)

Mathematica

seps[ptn_, fir_]:=If[Total[ptn]==1, {{fir}}, Join@@Table[Prepend[#, fir]&/@seps[MapAt[#-1&, ptn, fir], nex], {nex, Select[DeleteCases[Range[Length[ptn]], fir], ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0, {{}}, Join@@(seps[ptn, #]&/@Range[Length[ptn]])];
Table[Sum[Length[seps[y]], {y, Select[IntegerPartitions[n], UnsameQ@@#&]}], {n, 0, 10}]

Prog

(PARI) lista(n)={my(x='x, t='t, g(i)=sum(j=1, i, binomial(i-1, i-j)*t^j/j!)); [abs(subst(serlaplace(p), t, -1))|p<-Vec(prod(k=1, (n+1)\2, 1+O(x*x^n)+g(k)*x^k))]} \\ Christian Sievers, Jan 28 2026

Crossrefs

For any multiplicities we have A005649.

For weakly instead of strictly decreasing multiplicities we have A321688.

A003242 and A335452 count anti-runs, ranks A333489.

A005651 counts ordered set partitions with weakly decreasing sizes, strict A007837.

A032020 counts strict anti-run compositions.

A325534 counts separable multisets, ranks A335433.

A325535 counts inseparable multisets, ranks A335448.

A336103 counts normal separable multisets, inseparable A336102.

A386583 counts separable partitions by length, inseparable A386584.

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

A386586 counts partitions of inseparable type by length, sums A025065, ranks A335126.

A386633 counts separable set partitions, row sums of A386635.

A386634 counts inseparable set partitions, row sums of A386636.

Cf. A000670, A019472, A106351, A238130, A335125, A335516, A386580, A386581.

Sequence in context: A076587 A266999 A166204 * A364190 A329649 A040094

Adjacent sequences: A386573 A386574 A386575 * A386577 A386578 A386579

Keyword

nonn

Author

Gus Wiseman, Aug 03 2025

Extensions

a(16) onward from Christian Sievers, Jan 28 2026

Status

approved