A351008 - OEIS

A351008

Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093

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OFFSET

0,6

COMMENTS

Write the series in the g.f. given below as Sum_{k >= 0} q^(1 + 3 + 5 + ... + 2*k-1 + 2*k)/Product_{i = 1..2*k} 1 - q^i. Since 1/Product_{i = 1..2*k} 1 - q^i is the g.f. for partitions with parts <= 2*k, we see that the k-th summand of the series is the g.f. for partitions with largest part 2*k in which every odd number less than 2*k appears at least once as a part. The partitions of this type are conjugate to (and hence equinumerous with) the partitions (y_1, y_2, ..., y_{2*k}) of even length 2*k having strict decrease y_i > y_(i+1) for all odd i < 2*k. - Peter Bala, Jan 02 2024

LINKS

Table of n, a(n) for n=0..55.

FORMULA

Conjecture: a(n+1) = A122129(n+1) - A122130(n). - Gus Wiseman, Feb 21 2022

G.f.: Sum_{n >= 0} q^(n*(n+2))/Product_{k = 1..2*n} 1 - q^k = 1 + q^3 + q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 4*q^8 + 5*q^9 + 6*q^10 + .... - Peter Bala, Jan 02 2024

EXAMPLE

The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):

21 31 32 42 43 53 54 64 65 75 76

41 51 52 62 63 73 74 84 85

61 71 72 82 83 93 94

3221 81 91 92 A2 A3

4221 4321 A1 B1 B2

5221 4331 4332 C1

5321 5331 5332

6221 5421 5431

6321 6331

7221 6421

7321

8221

a(10) = 6: the six partitions 64, 73, 82, 91, 4321 and 5221 listed above have conjugate partitions 222211, 2221111, 22111111, 211111111, 4321 and 43111, These are the partitions of 10 with largest part L even and such that every odd number less than L appears at least once as a part. - Peter Bala, Jan 02 2024

MAPLE

series(add(q^(n*(n+2))/mul(1 - q^k, k = 1..2*n), n = 0..10), q, 141):

seq(coeftayl(%, q = 0, n), n = 0..140); # Peter Bala, Jan 03 2025

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]

CROSSREFS

The version for equal instead of unequal is A035363.

The alternately equal and unequal version is A035457, any length A351005.

This is the even-length case of A122129, opposite A122135.

The odd-length version appears to be A122130.

The alternately unequal and equal version is A351007, any length A351006.